## The human body converts internal chemical energy into work and heat at rates of 60 to 125W (called the basal metabolic rate). This energy comes from food and is usually measured in kilocalories [1 kcal = 4.186 kJ]. (Note that the nutritional ‘Calorie’ listed on packaged food actually equals 1 kilocalorie). How many kilocalories of food energy does a person with a metoblic rate of 103.0 W require per day? Enter the numerical answer without units.

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## Which of the following is true regarding a woman’s body cells? Question 2 options: Women have one X and one Y chromosome. Women have two Z chromosomes. Women have two Y chromosomes. Women have two X chromosomes.

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## The plate in figure is uniform and weigh 300 Ib. It is supported in a horizontal position by a hinge in the middle of CD and by a cable at point B. a) Draw a complete free body diagram for the plate . b) Determine the tension in the cable .

## The 300 kg mass, m, is supported at point C by cables CA and CB. A 200 N force, F, is also applied to the supporting hook at point C. The direction of F is given in the figure. The coordinates of points A, B, and C are also shown. . For all answers, use the x,y coordinate system shown. a. Draw the fiee body diagram (FBD) of the supporting hook at C. This FBD must I be a new figure; do not do it on top of the figure shown, (10 points). b. Write the cable force vector, Tcq, in Cartesian vector form, where the magnitude of the force vector is left as a variable (10 points). c. Write the cable force vector TCB, in Cartesian vector form, where the magnitude of the force vector is left as a variable (1 0 points). d. Determine the unit vector, ec~f,io m point C to B (10 points). e. Using the vector dot product, calculate the angle, a, between the two cables (1 0 points). f. Write (but do not solve) the two equations needed to calculate the tensions in the

## Extra Credit Due: 11:59pm on Thursday, May 15, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy A Man Running to Catch a Bus A man is running at speed (much less than the speed of light) to catch a bus already at a stop. At , when he is a distance from the door to the bus, the bus starts moving with the positive acceleration . Use a coordinate system with at the door of the stopped bus. Part A What is , the position of the man as a function of time? Answer symbolically in terms of the variables , , and . Hint 1. Which equation should you use for the man’s speed? Because the man’s speed is constant, you may use . ANSWER: c t = 0 b a x = 0 xman(t) b c t x(t) = x(0) + vt xman(t) = −b + ct Correct Part B What is , the position of the bus as a function of time? Answer symbolically in terms of and . Hint 1. Which equation should you use for the bus’s acceleration? Because the bus has constant acceleration, you may use . Recall that . ANSWER: Correct Part C What condition is necessary for the man to catch the bus? Assume he catches it at time . Hint 1. How to approach this problem If the man is to catch the bus, then at some moment in time , the man must arrive at the position of the door of the bus. How would you express this condition mathematically? ANSWER: xbus(t) a t x(t) = x(0) + v(0)t + (1/2)at2 vbus(0) = 0 xbus = 1 a 2 t2 tcatch tcatch Typesetting math: 15% Correct Part D Inserting the formulas you found for and into the condition , you obtain the following: , or . Intuitively, the man will not catch the bus unless he is running fast enough. In mathematical terms, there is a constraint on the man’s speed so that the equation above gives a solution for that is a real positive number. Find , the minimum value of for which the man will catch the bus. Express the minimum value for the man’s speed in terms of and . Hint 1. Consider the discriminant Use the quadratic equation to solve: . What is the discriminant (the part under the radical) of the solution for ? xman(tcatch) > xbus(tcatch) xman(tcatch) = xbus(tcatch) xman(tcatch) < xbus(tcatch) c = a tcatch xman(t) xbus(t) xman(tcatch) = xbus(tcatch) −b+ct = a catch 1 2 t2 catch 1 a −c +b = 0 2 t2 catch tcatch c tcatch cmin c a b 1 a − c + b = 0 2 t2 catch tcatch tcatch Typesetting math: 15% Hint 1. The quadratic formula Recall: If then ANSWER: Hint 2. What is the constraint? To get a real value for , the discriminant must be greater then or equal to zero. This condition yields a constraint that exceed . ANSWER: Correct Part E Assume that the man misses getting aboard when he first meets up with the bus. Does he get a second chance if he continues to run at the constant speed ? Hint 1. What is the general quadratic equation? The general quadratic equation is , where , \texttip{B}{B}, and \texttip{C}{C} are constants. Depending on the value of the discriminant, \Delta = c^2-2ab, the equation may have Ax2 + Bx + C = 0 x = −B±B2−4AC 2A = cc − 2ab tcatch c cmin cmin = (2ab) −−−− c > cmin Ax2 + Bx + C = 0 A Typesetting math: 15% two real valued solutions 1. if \Delta > 0, 2. one real valued solution if \Delta = 0, or 3. two complex valued solutions if \Delta < 0. In this case, every real valued solution corresponds to a time at which the man is at the same position as the door of the bus. ANSWER: Correct Adding and Subtracting Vectors Conceptual Question Six vectors (A to F) have the magnitudes and directions indicated in the figure. Part A No; there is no chance he is going to get aboard. Yes; he will get a second chance Typesetting math: 15% Which two vectors, when added, will have the largest (positive) x component? Hint 1. Largest x component The two vectors with the largest x components will, when combined, give the resultant with the largest x component. Keep in mind that positive x components are larger than negative x components. ANSWER: Correct Part B Which two vectors, when added, will have the largest (positive) y component? Hint 1. Largest y component The two vectors with the largest y components will, when combined, give the resultant with the largest y component. Keep in mind that positive y components are larger than negative y components. ANSWER: C and E E and F A and F C and D B and D Typesetting math: 15% Correct Part C Which two vectors, when subtracted (i.e., when one vector is subtracted from the other), will have the largest magnitude? Hint 1. Subtracting vectors To subtract two vectors, add a vector with the same magnitude but opposite direction of one of the vectors to the other vector. ANSWER: Correct Tactics Box 3.1 Determining the Components of a Vector Learning Goal: C and D A and F E and F A and B E and D A and F A and E D and B C and D E and F Typesetting math: 15% To practice Tactics Box 3.1 Determining the Components of a Vector. When a vector \texttip{\vec{A}}{A_vec} is decomposed into component vectors \texttip{\vec{A}_{\mit x}}{A_vec_x} and \texttip{\vec{A}_{\mit y}}{A_vec_y} parallel to the coordinate axes, we can describe each component vector with a single number (a scalar) called the component. This tactics box describes how to determine the x component and y component of vector \texttip{\vec{A}}{A_vec}, denoted \texttip{A_{\mit x}}{A_x} and \texttip{A_{\mit y}}{A_y}. TACTICS BOX 3.1 Determining the components of a vector The absolute value |A_x| of the x component \texttip{A_{\mit x}}{A_x} is the magnitude of the component vector \texttip{\vec{A}_{\1. mit x}}{A_vec_x}. The sign of \texttip{A_{\mit x}}{A_x} is positive if \texttip{\vec{A}_{\mit x}}{A_vec_x} points in the positive x direction; it is negative if \texttip{\vec{A}_{\mit x}}{A_vec_x} points in the negative x direction. 2. 3. The y component \texttip{A_{\mit y}}{A_y} is determined similarly. Part A What is the magnitude of the component vector \texttip{\vec{A}_{\mit x}}{A_vec_x} shown in the figure? Express your answer in meters to one significant figure. ANSWER: Correct |A_x| = 5 \rm m Typesetting math: 15% Part B What is the sign of the y component \texttip{A_{\mit y}}{A_y} of vector \texttip{\vec{A}}{A_vec} shown in the figure? ANSWER: Correct Part C Now, combine the information given in the tactics box above to find the x and y components, \texttip{B_{\mit x}}{B_x} and \texttip{B_{\mit y}}{B_y}, of vector \texttip{\vec{B}}{B_vec} shown in the figure. Express your answers, separated by a comma, in meters to one significant figure. positive negative Typesetting math: 15% ANSWER: Correct Conceptual Problem about Projectile Motion Learning Goal: To understand projectile motion by considering horizontal constant velocity motion and vertical constant acceleration motion independently. Projectile motion refers to the motion of unpowered objects (called projectiles) such as balls or stones moving near the surface of the earth under the influence of the earth's gravity alone. In this analysis we assume that air resistance can be neglected. An object undergoing projectile motion near the surface of the earth obeys the following rules: An object undergoing projectile motion travels horizontally at a constant rate. That is, the x component of its velocity, \texttip{v_{\mit x}}{1. v_x}, is constant. An object undergoing projectile motion moves vertically with a constant downward acceleration whose magnitude, denoted by \texttip{g}{g}, is equal to 9.80 \rm{m/s^2} near the surface of the earth. Hence, the y component of its velocity, \texttip{v_{\mit y}}{v_y}, changes continuously. 2. An object undergoing projectile motion will undergo the horizontal and vertical motions described above from the instant it is launched until the instant it strikes the ground again. Even though the horizontal and vertical motions can be treated independently, they are related by the fact that they occur for exactly the same amount of time, namely the time \texttip{t}{t} the projectile is in the air. 3. The figure shows the trajectory (i.e., the path) of a ball undergoing projectile motion over level ground. The time t_0 = 0\;\rm{s} corresponds to the moment just after the ball is launched from position x_0 = 0\;\rm{m} and y_0 = 0\;\rm{m}. Its launch velocity, also called the initial velocity, is \texttip{\vec{v}_{\rm 0}}{v_vec_0}. Two other points along the trajectory are indicated in the figure. One is the moment the ball reaches the peak of its trajectory, at time \texttip{t_{\rm 1}}{t_1} with velocity \texttip{\vec{v}_{\rm 1}}{v_1_vec}. Its position at this moment is denoted by (x_1, y_1) or (x_1, y_{\max}) since it is at its maximum \texttip{B_{\mit x}}{B_x}, \texttip{B_{\mit y}}{B_y} = -2,-5 \rm m, \rm m Typesetting math: 15% The other point, at time \texttip{t_{\rm 2}}{t_2} with velocity \texttip{\vec{v}_{\rm 2}}{v_2_vec}, corresponds to the moment just before the ball strikes the ground on the way back down. At this time its position is (x_2, y_2), also known as (x_{\max}, y_2) since it is at its maximum horizontal range. Projectile motion is symmetric about the peak, provided the object lands at the same vertical height from which is was launched, as is the case here. Hence y_2 = y_0 = 0\;\rm{m}. Part A How do the speeds \texttip{v_{\rm 0}}{v_0}, \texttip{v_{\rm 1}}{v_1}, and \texttip{v_{\rm 2}}{v_2} (at times \texttip{t_{\rm 0}}{t_0}, \texttip{t_{\rm 1}}{t_1}, and \texttip{t_{\rm 2}}{t_2}) compare? ANSWER: Correct Here \texttip{v_{\rm 0}}{v_0} equals \texttip{v_{\rm 2}}{v_2} by symmetry and both exceed \texttip{v_{\rm 1}}{v_1}. This is because \texttip{v_{\rm 0}}{v_0} and \texttip{v_{\rm 2}}{v_2} include vertical speed as well as the constant horizontal speed. Consider a diagram of the ball at time \texttip{t_{\rm 0}}{t_0}. Recall that \texttip{t_{\rm 0}}{t_0} refers to the instant just after the ball has been launched, so it is still at ground level (x_0 = y_0= 0\;\rm{m}). However, it is already moving with initial velocity \texttip{\vec{v}_{\rm 0}}{v_0_vec}, whose magnitude is v_0 = 30.0\;{\rm m/s} and direction is \theta = 60.0\;{\rm degrees} counterclockwise from the positive x direction. \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 1}}{v_1} = \texttip{v_{\rm 2}}{v_2} > 0 \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} = 0 \texttip{v_{\rm 0}}{v_0} = \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} > 0 \texttip{v_{\rm 0}}{v_0} > \texttip{v_{\rm 1}}{v_1} > \texttip{v_{\rm 2}}{v_2} > 0 \texttip{v_{\rm 0}}{v_0} > \texttip{v_{\rm 2}}{v_2} > \texttip{v_{\rm 1}}{v_1} = 0 Typesetting math: 15% Part B What are the values of the intial velocity vector components \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} (both in \rm{m/s}) as well as the acceleration vector components \texttip{a_{0,x}}{a_0, x} and \texttip{a_{0,y}}{a_0, y} (both in \rm{m/s^2})? Here the subscript 0 means “at time \texttip{t_{\rm 0}}{t_0}.” Hint 1. Determining components of a vector that is aligned with an axis If a vector points along a single axis direction, such as in the positive x direction, its x component will be its full magnitude, whereas its y component will be zero since the vector is perpendicular to the y direction. If the vector points in the negative x direction, its x component will be the negative of its full magnitude. Hint 2. Calculating the components of the initial velocity Notice that the vector \texttip{\vec{v}_{\rm 0}}{v_0_vec} points up and to the right. Since “up” is the positive y axis direction and “to the right” is the positive x axis direction, \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} will both be positive. As shown in the figure, \texttip{v_{0,x}}{v_0, x}, \texttip{v_{0,y}}{v_0, y}, and \texttip{v_{\rm 0}}{v_0} are three sides of a right triangle, one angle of which is \texttip{\theta }{theta}. Thus \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y} can be found using the definition of the sine and cosine functions given below. Recall that v_0 = 30.0\;\rm{m/s} and \theta = 60.0\;\rm{degrees} and note that \large{\sin(\theta) = \frac{\rm{length\;of\;opposite\;side}}{\rm{length\;of\;hypotenuse}}} \large{= \frac{v_{0, y}}{v_0}}, \large{\cos(\theta) = \frac{\rm{length\;of\;adjacent\;side}}{\rm{length\;of\;hypotenuse}}} \large{= \frac{v_{0, x}}{v_0}.} What are the values of \texttip{v_{0,x}}{v_0, x} and \texttip{v_{0,y}}{v_0, y}? Enter your answers numerically in meters per second separated by a comma. ANSWER: ANSWER: 15.0,26.0 \rm{m/s} Typesetting math: 15% Correct Also notice that at time \texttip{t_{\rm 2}}{t_2}, just before the ball lands, its velocity components are v_{2, x} = 15\;\rm{m/s} (the same as always) and v_{2, y} = – 26.0\;\rm{m/s} (the same size but opposite sign from \texttip{v_{0,y}}{v_0, y} by symmetry). The acceleration at time \texttip{t_{\rm 2}}{t_2} will have components (0, -9.80 \rm{m/s^2}), exactly the same as at \texttip{t_{\rm 0}}{t_0}, as required by Rule 2. The peak of the trajectory occurs at time \texttip{t_{\rm 1}}{t_1}. This is the point where the ball reaches its maximum height \texttip{y_{\rm max}}{y_max}. At the peak the ball switches from moving up to moving down, even as it continues to travel horizontally at a constant rate. Part C What are the values of the velocity vector components \texttip{v_{1,x}}{v_1, x} and \texttip{v_{1,y}}{v_1, y} (both in \rm{m/s}) as well as the acceleration vector components \texttip{a_{1,x}}{a_1, x} and \texttip{a_{1,y}}{a_1, y} (both in \rm{m/s^2})? Here the subscript 1 means that these are all at time \texttip{t_{\rm 1}}{t_1}. ANSWER: 30.0, 0, 0, 0 0, 30.0, 0, 0 15.0, 26.0, 0, 0 30.0, 0, 0, -9.80 0, 30.0, 0, -9.80 15.0, 26.0, 0, -9.80 15.0, 26.0, 0, +9.80 Typesetting math: 15% Correct At the peak of its trajectory the ball continues traveling horizontally at a constant rate. However, at this moment it stops moving up and is about to move back down. This constitutes a downward-directed change in velocity, so the ball is accelerating downward even at the peak. The flight time refers to the total amount of time the ball is in the air, from just after it is launched (\texttip{t_{\rm 0}}{t_0}) until just before it lands (\texttip{t_{\rm 2}}{t_2}). Hence the flight time can be calculated as t_2 – t_0, or just \texttip{t_{\rm 2}}{t_2} in this particular situation since t_0 = 0. Because the ball lands at the same height from which it was launched, by symmetry it spends half its flight time traveling up to the peak and the other half traveling back down. The flight time is determined by the initial vertical component of the velocity and by the acceleration. The flight time does not depend on whether the object is moving horizontally while it is in the air. Part D If a second ball were dropped from rest from height \texttip{y_{\rm max}}{y_max}, how long would it take to reach the ground? Ignore air resistance. Check all that apply. Hint 1. Kicking a ball of cliff; a related problem Consider two balls, one of which is dropped from rest off the edge of a cliff at the same moment that the other is kicked horizontally off the edge of the cliff. Which ball reaches the level ground at the base of the cliff first? Ignore air resistance. Hint 1. Comparing position, velocity, and acceleration of the two balls Both balls start at the same height and have the same initial y velocity (v_{0,y} = 0) as well as the same acceleration (\vec a = g downward). They differ only in their x velocity (one is 0, 0, 0, 0 0, 0, 0, -9.80 15.0, 0, 0, 0 15.0, 0, 0, -9.80 0, 26.0, 0, 0 0, 26.0, 0, -9.80 15.0, 26.0, 0, 0 15.0, 26.0, 0, -9.80 Typesetting math: 15% zero, the other nonzero). This difference will affect their x motion but not their y motion. ANSWER: ANSWER: Correct In projectile motion over level ground, it takes an object just as long to rise from the ground to the peak as it takes for it to fall from the peak back to the ground. The range \texttip{R}{R} of the ball refers to how far it moves horizontally, from just after it is launched until just before it lands. Range is defined as x_2 – x_0, or just \texttip{x_{\rm 2}}{x_2} in this particular situation since x_0 = 0. Range can be calculated as the product of the flight time \texttip{t_{\rm 2}}{t_2} and the x component of the velocity \texttip{v_{\mit x}}{v_x} (which is the same at all times, so v_x = v_{0,x}). The value of \texttip{v_{\mit x}}{v_x} can be found from the launch speed \texttip{v_{\rm 0}}{v_0} and the launch angle \texttip{\theta }{theta} using trigonometric functions, as was done in Part B. The flight time is related to the initial y component of the velocity, which may also be found from \texttip{v_{\rm 0}}{v_0} and \texttip{\theta }{theta} using trig functions. The following equations may be useful in solving projectile motion problems, but these equations apply only to a projectile launched over level ground from position (x_0 = y_0 = 0) at time t_0 = 0 with initial speed \texttip{v_{\rm 0}}{v_0} and launch angle \texttip{\theta }{theta} measured from the horizontal. As was the case above, \texttip{t_{\rm 2}}{t_2} refers to the flight time and \texttip{R}{R} refers to the range of the projectile. flight time: \large{t_2 = \frac{2 v_{0, y}}{g} = \frac{2 v_0 \sin(\theta)}{g}} range: \large{R = v_x t_2 = \frac{v_0^2 \sin(2\theta)}{g}} The ball that falls straight down strikes the ground first. The ball that was kicked so it moves horizontally as it falls strikes the ground first. Both balls strike the ground at the same time. \texttip{t_{\rm 0}}{t_0} t_1 – t_0 \texttip{t_{\rm 2}}{t_2} t_2 – t_1 \large{\frac{t_2 – t_0}{2}} Typesetting math: 15% In general, a high launch angle yields a long flight time but a small horizontal speed and hence little range. A low launch angle gives a larger horizontal speed, but less flight time in which to accumulate range. The launch angle that achieves the maximum range for projectile motion over level ground is 45 degrees. Part E Which of the following changes would increase the range of the ball shown in the original figure? Check all that apply. ANSWER: Correct A solid understanding of the concepts of projectile motion will take you far, including giving you additional insight into the solution of projectile motion problems numerically. Even when the object does not land at the same height from which is was launched, the rules given in the introduction will still be useful. Recall that air resistance is assumed to be negligible here, so this projectile motion analysis may not be the best choice for describing things like frisbees or feathers, whose motion is strongly influenced by air. The value of the gravitational free-fall acceleration \texttip{g}{g} is also assumed to be constant, which may not be appropriate for objects that move vertically through distances of hundreds of kilometers, like rockets or missiles. However, for problems that involve relatively dense projectiles moving close to the surface of the earth, these assumptions are reasonable. A World-Class Sprinter World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude 15 \;{\rm m}/{\rm s}^{2}. Part A How much horizontal force \texttip{F}{F} must a sprinter of mass 64{\rm kg} exert on the starting blocks to produce this acceleration? Express your answer in newtons using two significant figures. Increase \texttip{v_{\rm 0}}{v_0} above 30 \rm{m/s}. Reduce \texttip{v_{\rm 0}}{v_0} below 30 \rm{m/s}. Reduce \texttip{\theta }{theta} from 60 \rm{degrees} to 45 \rm{degrees}. Reduce \texttip{\theta }{theta} from 60 \rm{degrees} to less than 30 \rm{degrees}. Increase \texttip{\theta }{theta} from 60 \rm{degrees} up toward 90 \rm{degrees}. Typesetting math: 15% Hint 1. Newton’s 2nd law of motion According to Newton’s 2nd law of motion, if a net external force \texttip{F_{\rm net}}{F_net} acts on a body, the body accelerates, and the net force is equal to the mass \texttip{m}{m} of the body times the acceleration \texttip{a}{a} of the body: F_{\rm net} = ma. ANSWER: Co

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## 1) Can two different forces, acting through the same point, produce the same torque on an object? Answer: Yes, as long as the component of the force perpendicular to the line joining the axis to the force is the same for both forces. 2) If you stand with your back towards a wall and your heels touching the wall, you cannot lean over to touch your toes. Why? Answer: As you bend over your center of gravity moves forward and eventually is beyond the area of the floor in touch with your feet. This does not happen when you do it away from the wall because part of your body moves back and the center of mass remains over your feet. 3) Two equal forces are applied to a door at the doorknob. The first force is applied perpendicular to the door; the second force is applied at 30° to the plane of the door. Which force exerts the greater torque? A) the first applied perpendicular to the door B) the second applied at an angle C) both exert equal non-zero torques D) both exert zero torques E) Additional information is needed. 4) A heavy boy and a lightweight girl are balanced on a massless seesaw. If they both move forward so that they are one-half their original distance from the pivot point, what will happen to the seesaw? A) It is impossible to say without knowing the masses. B) It is impossible to say without knowing the distances. C) The side the boy is sitting on will tilt downward. D) Nothing, the seesaw will still be balanced. E) The side the girl is sitting on will tilt downward. 5) A figure skater is spinning slowly with arms outstretched. She brings her arms in close to her body and her angular speed increases dramatically. The speed increase is a demonstration of A) conservation of energy: her moment of inertia is decreased, and so her angular speed must increase to conserve energy. B) conservation of angular momentum: her moment of inertia is decreased, and so her angular speed must increase to conserve angular momentum. C) Newton’s second law for rotational motion: she exerts a torque and so her angular speed increases. D) This has nothing to do with mechanics, it is simply a result of her natural ability to perform. 6) A girl weighing 450. N sits on one end of a seesaw that is 3.0 m long and is pivoted 1.3 m from the girl. If the seesaw is just balanced when a boy sits at the opposite end, what is his weight? Neglect the weight of the seesaw. 7) An 82.0 kg painter stands on a long horizontal board 1.55 m from one end. The 15.5 kg board is 5.50 m long. The board is supported at each end. (a) What is the total force provided by both supports? (b) With what force does the support, closest to the painter, push upward? FIGURE 11-4 8) The mobile shown in Figure 11-4 is perfectly balanced. What must be the masses of m1, m2, and m3?

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## Physics 2010 Sid Rudolph Fall 2009 MIDTERM 4 REVIEW Problems marked with an asterisk (*) are for the final. Solutions are on the course web page. 1. A. The drawing to the right shows glass tubing, a rubber bulb and two bottles. Is the situation you see possible? If so, carefully describe what has taken place in order to produce the situation depicted. B. The picture depicts three glass vessels, each filled with a liquid. The liquids each have different densities, and DA > DB > DC. In vessel B sits an unknown block halfway to the bottom and completely submerged. 1. _______ In which vessel would the block sit on the bottom? 2. _______ In which vessel would the block float on the top? 3. _______ In which vessel would the block feel the smallest buoyant force? 4. _______ In which vessels are buoyant forces on the block are the same? 5. _______ Assume the coefficient of volume expansion for the liquid in B and the block are $B > $block. If the temperature of vessel B with the block is raised, would block B rise to the surface, sink to the bottom, or remain where it is? 2. A circular tank with a 1.50 m radius is filled with two fluids, a 4.00 m layer of water and a 3.00 m layer of oil. Use Doil = 8.24 × 10 kg/m and Dwater = 1.00 × 10 kg/m , and Datm = 1.01 × 10 N/m . 2 3 3 3 5 2 A. What are the gauge and absolute pressures 1.00 m above the bottom of the tank? B. A block of material in the shape of a cube (m = 100 kg and side length = 42.0 cm) is released at the top of the oil layer. Where does the block come to rest? Justify your answer. If it comes to rest between two layers, specify which layers and what portion of the block sits in each layer. [Note: Vcube = a ]3 C. A small 1.00 cm radius opening is made in the side of the tank 0.500 m up from its base (block was removed). What volume of water drains from the tank in 10.0 s? (b) (a) 3. A tube is inserted into a vein in the wrist of a patient in a reclined position on a hospital bed. The heart is vertically 25.0 cm above the position of the wrist where the tube is inserted. Take DBLOOD = 1.06 × 103 kg/m3. The gauge venous blood pressure at the level of the heart is 6.16 × 103 N/m2. Assume blood behaves as an ideal nonviscous fluid. A. What is the gauge venous blood pressure at the position of the wrist? B. The tube coming from the wrist is connected to a bottle of whole blood the patient needs in a transfusion. See above figure (b). What is the minimum height above the level of the heart at which the bottle must be held to deliver the blood to the patient? C. Suppose the bottle of blood is held 1.00 m above the level of the heart. Assume the tube inserted in the wrist has a diameter of 2.80 mm. What is the velocity, v, and flow rate of blood as it enters the wrist. You may also assume the rate at which the blood level in the bottle drops is very small. The answer you get here is a substantial overstatement. Blood is not really a non-viscous fluid. 4. A 0.500 kg block is attached to a horizontal spring and oscillates back and forth on a frictionless surface with a frequency of f = 3.00 hz. The amplitude of this motion is 6.00 × 10 m. Assume to = 0 and is the instant the block is -2 at the equilibrium position moving to the left. A. Write expressions x(t) = !A sin (Tt) and v(t) = !AT cos (Tt) filling in the values of A and T. B. What is the total mechanical energy (METOT) of the block-spring system? C. Suppose the block, at the moment it reaches its maximum velocity to the left splits in half with only one of the halves remaining attached to the spring. What are the amplitude and frequency of the resulting oscillations? D. Suppose, instead of splitting at the position of maximum velocity to the left, the block now splits when it is at the extreme position in the left. What are the amplitude and frequency of the resulting motion? E. Describe in words what would happen to the period of oscillation if a second block identical to the first block were dropped on the first block at either of its extreme positions. 5. A. A spring has one end attached to a wall and the other end attached to two identical masses, mA and mB. The system is set into oscillation on a frictionless surface with amplitude A. See figure. When the system is momentarily at rest at x = -A whatever it is that holds mA to mB fails; and later in the motion mB moves away from mA to the right. 1. Location where the acceleration of mA is maximum and negative. 2. Location where the KE of mA is maximum. The next few questions ask you to compare the behavior of the mass-spring system after and before mB detached. Energy considerations are most useful here. 3. The amplitude of the mass-spring oscillation has (increased, decreased, not changed) after mB detaches. 4. The frequency of the mass-spring oscillation has (increased, decreased, stayed the same) after mB detaches. 5. The maximum speed of mA has (increased, decreased, stayed the same) after mB detaches. 6. The period of oscillation of the mass-spring system has (increased, decreased, stayed the same) after mB detaches. 7. The fraction of the total mechanical energy of the entire spring-2 mass system carried away with mB after mB detaches is B. A spherical object is completely immersed in a liquid and is neutrally buoyant some distance above the bottom of the vessel. See figure. The upper surface of the liquid is open to the earth’s atmosphere. 1. How is the density of the fluid related to the density of the spherical object? 2. Assume the fluid and object are incompressible. In addition, the $sphere (coefficient of volume expansion) > $liquid. For the following items below, indicate whether the object sinks to the bottom, rises to the surface, or does nothing based on the changes described. a. Atmospheric pressure drops by 20%. b. Salt is dissolved in the liquid in the same way fresh water is turned into salt water. c. The entire apparatus is warmed 10oC (liquid and object are both warmed). d. The entire apparatus is transported to the surface of the moon. (gmoon = 1.6 m/s , PATM = 0 on moon) 2 e. 100 cm3 of the liquid is removed from the top. The object is still initially submerged. 6. A. A mass m is attached to a spring and oscillating on a frictionless, horizontal surface. See figure. At the instant the mass passes the equilibrium position moving to the right, half the mass detaches from the other half. The oscillating system is now the spring and half the original mass with the detached mass moving off to the right with constant velocity. Relative to the original spring-mass system, the new spring-mass system with half the mass oscillates with … In the spaces provided below, enter the words larger, smaller or the same that best completes the above sentence.. 1. amplitude 2. period 3. frequency 4. maximum velocity 5. mechanical energy B. A solid cylinder is floating at the interface between water and oil with 3/4 of the cylinder in the water region and 1/4 of the cylinder in the oil region. See figure. Select the item in the parenthesis that best fits the statement. 1. The item (oil, water, and/or cylinder) with the largest density. 2. The item (oil, water, and/or cylinder) with the smallest density. 3. The weight of the cylinder (is equal to, greater than or less than) the total buoyant force it feels. 4. The density of the cylinder (is equal to, less than, or greater than) the density of water. rC. Three thermometers in different settings record temperatures T1 = 1000°F, T2 = 1000°C, and T3 = 1000 K. In the space below select T1, T2 or T3, that best fits the statement. 1. The thermometer in the hottest environment. 2. The thermometer in the coolest environment. 3. The thermometer reading a temperature 900° above the boiling point of water. 7. An oil tanker in the shape of a rectangular solid is filled with oil (Doil = 880 kg/m ). The flat bottom of the 3 hull is 48.0 m wide and sits 26.0 m below the surface of the surrounding water. Inside the hull the oil is stored to a depth of 24.0 m. The length of the tanker, assumed filled with oil along the entire length, is 280 m. View from Rear View from Side Note: Dsalt water = 1.015 × 10 kg/m ; Vrectangular solid = length × width × height. 3 3 A. At the bottom of the hull, what is the water pressure on the outside and the oil pressure on the inside of the horizontal bottom part of the hull? Assume the Po above the oil is the same as the Po above the water and its value is Po = 1.01 × 10 N/m . 5 2 B. If you did part A correctly you determined that the water pressure on the horizontal bottom part of the hull is larger than the oil pressure there. Explain why this MUST be the case. C. What buoyant force does the tanker feel? D. What is the weight of the tanker, excluding the weight of the oil in the hull? 8. A. Water is poured into a tall glass cylinder until it reaches a height of 24.0 cm above the bottom of the cylinder. Next, olive oil (Doil = 920 kg/m ) is very carefully added until the total amount of 3 fluid reaches 48.0 cm above the bottom of the cylinder. Olive oil and water do not mix. See figure. Take Dwater = 1.00 × 10 kg/m and Patm = 1.01 × 10 N/m . 3 3 5 2 1. Indicate on the drawing which layer is water and which is olive oil. 2. What is the gauge pressure 10.0 cm below the top of the upper fluid layer in the cylinder. 3. What is the gauge pressure on the bottom of the cylinder? 4. If the cylinder is in the shape of a right circular cylinder with radius of 3.60 cm, what force is exerted on the bottom of the cylinder? B. A 0.200 kg mass is hung from a massless spring. At equilibrium, the spring stretched 28.0 cm below its unstretched length. This mass is now replaced with a 0.500 kg mass. The 0.500 kg mass is lowered to the original equilibrium position of the 0.200 kg mass and suddenly released producing vertical SHM. 1. What is the spring constant for this spring? 2. What is the period of oscillation for the 0.500 kg/spring system? 3. What is the amplitude of this oscillation? r9. The drawing shows a possible design for a thermostat. It consists of an aluminum rod whose length is 5.00 cm at 20.0°C. The thermostat switches an air conditioner when the end of the rod just touches the contact. The position of the contact can be changed with an adjustment screw. What is the size of the spacing such that the air conditioner turns on at 27.0°C. This is not a very practical device. Take “al = 2.3 × 10 /°C. -5 r10. The following is an effective technique for determining the temperature TF inside a furnace. Inside the furnace is 100 gm of molten (i.e., in a liquid state) lead (Pb). The lead is dropped into an aluminum calorimeter containing 200 gm water both at an initial temperature of 10.0°C. After equilibrium is reached, the temperature reads 21.8°C. Assumptions: (1) No water is vaporized; (2) no heat is lost to or gained from the environment; and (3) the specific heat for the lead is the same whether the lead is a solid or a liquid. DATA TABLE LEAD CALORIMETER WATER mPb = 100 gm mAl = 150 gm mW = 200 gm CPb = 0.0305 cal/gm°C CAl = 0.215 cal/gm°C CW = 1.0 cal/gm°C LF = 6.0 ca./gm (heat of fusion) Tinit = 10.0°C Tinit = 10.0°C MPPb = 327°C (melting point) TF = unknown Tequilibrium = 21.8°C A. In words, describe the distinct steps in the cooling of lead. B. How many calories of heat are absorbed by the calorimeter and the water it contains to reach 21.8°C? C. How many calories are lost by the lead in cooling from TF to the final equilibrium temperature of 21.8°C? D. What was the original furnace temperature? E. If the same amount of aluminum (CAl = 0.215 cal/gm°C and LM = 21.5 cal/gm) were used in the same furnace instead of lead, would the final equilibrium temperature be higher, less or the same as in the lead case? No calculation is needed to answer this. Please explain. r11. The length of aluminum cable between consecutive support towers carrying electricity to a large metropolitan area is 180.00 m on a hot August day when the temperature is 38°C. Use “(Al) = 24 × 10-6/°C. A. What is the length of the same section of aluminum cable on a very cold winter day when T = -24°C? B. If the same length of copper (” = 17 × 10-6/°C) cable (i.e., 180.00 m on the same hot August day) were used instead of aluminum, would the length of the copper cable be shorter, longer or the same as that of the aluminum on the same winter day as in (A)? Please explain your conclusion You do not have to do any calculations here. r12. You wish to make a cup of coffee with cream in a 0.250 kg mug (cmug = 900 J/kg°C) with 0.325 kg coffee (ccoffee = 4.18 × 10 J/kg°C) starting at 25.0°C and 0.010 kg cream (ccream = 3.80 × 10 J/kg°C) at 10.0°C. 3 3 You use a 50.0 W electric heater to bring the coffee, cream and mug to a final temperature of 90.0°C. How long must the coffee system be heated? Indicate clearly the assumptions you need to make. r13. A 75.0 kg patient is running a fever of 106°F and is given an alcohol rubdown to lower his body temperature. Take the specific heat of the human body to be Cbody = 3.48 × 10 J/kg°C, the heat of 3 evaporation of the rubbing alcohol to be Lv(alcohol) = 8.51 × 10 J/kg, and the density of the rubbing 5 alcohol to be 793 kg/m3. You may assume that all the heat removed from the fevered body goes into evaporating the alcohol, and that while the patient’s body is cooling, his metabolism adds no measurable heat. A. What quantity of heat must be removed from the body to lower its temperature to 99.0°F? B. What volume of rubbing alcohol is required? C. This is a qualitative question. Give an answer and explanation. Suppose you were told that the alcohol applied started at room temperature (. 70°F) and were given the specific heat for the alcohol. Thus, you now expect some of the body heat warming the alcohol to the temperature of the fever before evaporation occurs. How would this effect the result of the calculation in part (B)? r14. A 56.0 kg hypothermia victim is running a body temperature of 91.0°F. The victim is far away from any immediate medical treatment. Her friends decide to treat the hypothermia victim by placing the victim in a sleeping bag with one of her friends and use the heat from the friend to raise the victim’s body temperature. Take the specific heat of the human body to be Cbody = 3.48 × 10 J/kg°C. Assume that the sleeping bag acts 3 like a perfect calorimeter and also assume no heat is lost to or obtained from the sleeping bag. Finally, assume all the heat that warms the hypothermia victim comes from the basic metabolic heat produced by the body of the victim’s friend in the sleeping bag with her and that metabolism is rated at 2.00 × 106 cal/day, and that the victim’s metabolism is negligible. A. How much heat must be added to the victim’s body to get her temperature up to 98.0°F? B. How long must the victim remain in the sleeping bag with her friend to achieve this temperature change? C. This is a qualitative question. If the thermal characteristics of the sleeping bag are now taken into account, but still assuming no heat leaves or enters the sleeping bag, how will the answer to question (b) above be different? r15. A few years back a lawsuit was filed by a woman against McDonald’s because she scalded herself with a Styrofoam cup filled with coffee which she spilled on herself while driving. This question was spawned by that incredible legal action and represents a possible action taken by McDonald’s to insure cooler coffee. Suppose a typical cup of coffee sold by McDonald’s is basically 400 ml of hot water and when poured into the Styrofoam cup its temperature is 96.0°C. Take 1.00 ml to have a mass of 1.00 gm and = 4.19 kJ/kg°C. Neglect any heat lost to the cup and assume no heat is lost by the coffee to the environment. A. How much heat in joules must the coffee lose to bring its temperature to a drinkable 68.0°C? B. McDonald’s possible approach to lowering the temperature of the 96.0°C coffee to 68.0°C is to add a cube of ice initially at 0.0°C. (Take Lf = 334 kJ/kg.) What mass of ice has to be added to the coffee to reduce its initial temperature to the desired 68.0°C? r16. During this past Thanksgiving your instructor overdid it and consumed 3000 Cal of food and dessert. Remember 1.0 Cal = 4.19 x 10 J. For the questions below, as 3 sume no heat is lost to the environment. [Note: = 33.5 x 105 J/kg; = 4.19 x 103 J/kgoC] A. If all of this energy went into heating 65.0 kg water starting at 37.0oC (a mass approximately that of your instructor), what would be the final temperature of this water? B. Assume your instructor removes these overeating calories by running 10 kilometer races [note: 1.61 km = 1.00 mile]. Using the rule of thumb that 1 mile of jogging will require 100 Cal, what is the minimum number of races your instructor must run to consume the 3000 Cal in part A as exercise? C. The year before, your instructor was particularly gluttonous and consumed 5000 Cal. Assuming the same conditions of water mass (65.0 kg) and starting temperature (37.0oC) as in A, what is the final temperature of the water system, and if any water vaporizes to steam, how much? [Note: BP(H2O) = 100 C] o 17. Below is the position vs. time graph for the simple harmonic of a spring oscillation on a frictionless horizontal surface. Motion to the right is positive. 1. The earliest instant of time, including t0 = 0 at which the PEelastic is maximum. 2. The earliest instant of time at which the KE of the mass is a maximum and the mass is moving to the right. 3. The earliest instant of time at which the acceleration of the mass is maximum and positive. 4. The earliest instant of time at which the speed of the mass is zero. 18. A. A spring is attached to a post at the top of a 15.0° frictionless ramp. A 2.00 kg mass is attached to the spring and the mass is slowly allowed to stretch the spring to the equilibrium position of the mass-spring system, the spring stretches by 0.400 m See figure. The mass is now pulled an additional 10.0 cm and released. The mass-spring system executes simple harmonic motion. 1. What is the spring constant, k, of the spring. 2. What are the amplitude and period of oscillation of the mass-spring system? B. A solid, uniform cylinder is floating at the interface between water (Dwater = 1.00 × 103 kg/m ) and oil (Doil = 8.24 × 10 kg/m ) with 3/4 of the cylinder in the water region and 3 3 3 1/4 of the cylinder in the oil region. Assume the axis of the cylinder is perfectly vertical. See figure. 1. What is the density of the material out of which the cylinder is made? 2. Assume the upper surface of the oil region si open to the atmosphere (Datm = 1.01 × 10 N/m ) and the oil-water interface is 0.500 m below the 5 2 upper surface of the oil. Also assume the height of the cylinder is 10.0 cm. What is the gauge pressure on the bottom surface of the cylinder? Recall: Pgauge = P – PATM. 19. A. A mass m is attached to a spring and is oscillating on a frictionless horizontal surface (see figure). At the instant the mass is at an amplitude position a second identical mass is carefully placed on top of the original mass. The oscillating system is now the spring and the two identical masses. Relative to the original spring-single mass system, the new spring-2-mass system oscillates with a … In the spaces provided below, enter (I) for increased, (D) for decreased, or (R) remains unchanged, that best completes the above last sentence. 1. amplitude. 2. period. 3. frequency. 4. spring constant. 5. maximum speed. 6. mechanical energy. 7. maximum acceleration. B. Suppose you are asked about the absolute pressure at some depth h below the surface of a liquid. The top surface is exposed to the atmosphere on a sunny day in Salt Lake City. For each statement below in the spaces provided, enter I for increase, D for decrease, or R for remains the same, when accounting for what happens to the absolute pressure at the point you are observing. 1. More liquid is added so now the observation point is farther below the surface. 2. The fluid is now exchanged for a less dense fluid. The observation point is at same h. 3. The experiment is moved to New York City, which is at sea level, on a sunny day. 4. The fluid is now seen to be moving with some speed v past the observation point. 5. The observation point is moved closer to the surface of the liquid. 6. The air above the fluid is removed by a vacuum system. 7. The apparatus is moved to a laboratory on the surface of the moon. 20. A 3.00 kg mass is attached to a spring (k = 52.0 N/m) that is hanging vertically from a fixed support. The mass is moved to a position 0.800 m lower than the unstretched position of the end of the spring. The spring is then released and the mass-spring system executes SHM. Take the 0.800 m of the mass as the reference location for its gravitational PE. A. What is the equilibrium position of the mass-spring system? B. What is the amplitude of the SHM the mass-spring system executes? C. What is the period of the oscillation of this system? D. What is the total mechanical energy of the mass-spring system at the moment the mass is released? E. What are (i) the KE of the mass and (ii) the speed of the mass when the spring is at its equilibrium position? 21. A 38.0 kg block is moving back and forth on a frictionless horizontal surface between two springs. The spring on the right has a force constant kR = 2.50 × 10 N/m. When the block is between the two 3 springs its speed (v) is 1.82 m/s. See figure. A. If the block compresses the left spring to 5.62 cm beyond its uncompressed length, determine the value of kL. B. What is the maximum compression of the right spring when the mass interacts with it? C. What is the total time the spring on the right is compressed during a single event? 22. Two identical containers are connected at the bottom via a tube of negligible volume and a valve which is closed. Both containers are filled initially to the same height of 1.00 m, one with chloroform (DC = 1530 kg/m ) in the left chamber and the other 3 with mercury in the right chamber (DHg = 1.36 × 10 kg/m ). 4 3 Sitting on top of each identical circular container is a massless plate that can slide up or down without friction and without allowing any fluid to leak past. The radius of the circular plate is 12.0 cm. The valve is now opened. A. What volume of mercury drains into the chloroform container? (Note: Vcyl = Br h) 2 B. What mass must be placed on the plate on the chloroform side to force all the mercury, but none of the chloroform, back to the mercury chamber? 23. A 12.0 kg mass M is attached to a cord that is wrapped around a wheel in the shape of a uniform disk of radius r = 12.0 cm and mass m = 10.0 kg. The block starts from rest and accelerates down the frictionless incline with constant acceleration. Assume the disk axle is frictionless. Note: Idisk = 1/2 mr . 2 A. Use energy methods to find the velocity of the block after it has moved 2.00 m down the incline. B. What is the constant acceleration of the block and the angular acceleration of the wheel? C. How many revolutions does the wheel turn for the distance the block travels in (A)? D. If the uniform disk were replaced by a uniform sphere with the same r and m of the disk, would the acceleration of the block attached to the sphere be larger, smaller, or the same as that for the block attached to the disk? Note: Isphere = 2/5 mr . 2 24. A pulley is in the shape of a uniform disk of mass m = 5.00 kg and radius r = 6.40 cm. The pulley can rotate without friction about an axis through the center of mass. A massless cord is wrapped around the pulley and connected to a 1.80 kg mass. The 1.80 kg mass is released from rest and falls 1.50 m. See figure. Note: Idisk = 1/2 mr . 2 A. Use energy methods to determine the speed of the block after falling 1.50 m. B. What is the constant acceleration of the block and the angular acceleration of the wheel? C. How many revolutions does the pulley disk turn for the distance the block travels in (A)? D Suppose the disk were replaced by a uniform sphere with the same r and m of the disk. Would the acceleration of the block attached to the sphere be larger, smaller, or the same as that for the block attached to the the disk? Note: Isphere 2/5 mr . 2 26. A 700.0 N fisherman is walking toward the edge of a 200 N plank as shown. He has placed a can of worms weighing 75.0 N on the left side of the plank as indicated in the drawing. The plank is the horizontal section in the drawing. A. Identify all the forces the plank feels before it begins to tip. Draw a free body diagram. B. As the fisherman nears the point on the plank where it begins to tip, how do the upward forces the supports exert on the plank change. C. How far a distance, as measured from the center of the right support, can he walk before the plank begins to tip? 26. A 75.0 kg sign hangs from a 4.80 m uniform horizontal rod whose mass is 120 kg. The rod is supported by a cable that makes an angle of 53° with the rod. he sign hangs 3.60 m out along the rod. A. What is the tension in the cable? B. What are the forces PPv and PPH exerted by the wall on the left end of the rod? 27. A 1.00 × 104 N great white shark is hanging by a cable attached to a 4.00 m massless rod that can pivot at its base. See figure. A. Determine the tension in the cable supporting the upper end of the rod. See figure. B. Determine the force (a vector quantity) exerted on the base of the rod. Suggestion: Find this force by first evaluating the separate components of the force. See figure. 28. A 6.00 m uniform beam extends horizontally from a hinge fixed on a wall on the left. A cable is attached to the right end of the beam. The cable makes an angle of 30.0° with respect to the horizontal and the right end of the cable is fixed to a wall on the right. At the right end of the cable hangs a 140.0 kg mass. The mass of the beam is 240.0 kg. See figure. A. Find the tension in the cable. B. Find the vertical and horizontal forces the hinge exerts on the left end of the beam. 29 A. The blades of a “Cuisinart” blender when run at the “mix” level, start from rest and reach 2.00 × 103 rpm (revolutions per minute) in 1.60 s. The edges of the blades are 3.10 cm from the center of the circle about which they rotate. 1. What is the angular acceleration of the blades in rad/s2 while they are accelerating? 2. Through how many rotations did the blades travel in that 1.60 s? 3. If the blades have a moment of inertia of 5.00 × 10-5 kg m2, what net torque did the blades feel while accelerating? B. A 7.50 × 10 N 4 shipping crate is hanging by a cable attached to a uniform 1.20 × 104 N steel beam that can pivot at its base. A second cable supports the beam and is attached to a wall. See figure. 1. Determine the tension T in the upper cable. 2. Determine the magnitude of the force exerted on the beam at its base. See drawing. 30. The drawing shows a uniform ladder of length L and weight 220 N. The ladder is sitting at an angle of 30° above the horizontal resting on the corner of a concrete wall at a point that is one-fourth of the way from the end of the ladder. A 640 N construction worker is standing on the ladder one-third of the way up from the end of the ladder which is resting on the ground. Assume the corner of the wall on which the ladder rests exerts only a normal force on the ladder at the point where there is contact. A. What is the magnitude of the normal force the wall exerts on the ladder? B. Find the magnitude of both the normal force the ground exerts on the left end of the ladder and the static frictional force the ground exerts on the left end of the ladder. 31. A. A solid, right circular cylinder (radius = 0.150 m, height = 0.120 m) has a mass m. The cylinder is floating in a tank in the interface between two liquids that do not mix: water on the bottom and oil above. One-third of the cylinder is in the oil layer (Doil = 725 kg/m ) 3 and two-thirds in the water layer (Dwater = 1.00 × 10 kg/m ). See 3 3 drawing. Note: V(circular cylinder) = B r2 h. 1. Find the mass of the cylinder. 2. With the cylinder present, take the thickness of the oil layer to be 0.200 m and the thickness of the water layer to be 0.300 m. What is the gauge pressure at the bottom of the tank? Assume the top of the oil layer is exposed to the atmosphere. B. A block rests on a frictionless horizontal surface and is attached to a spring. When set into simple harmonic motion, the block oscillates back and forth with an angular frequency of T = 7.52 rad/s. The drawing indicates the position of the block when the spring is unstretched. That position is labeled “x = 0 m” in the drawing. The drawing also shows a small bottle whose left edge is located at Xb = 0.0900 m. The block is now pulled to the right, stretching the spring by Xs = 0.0343 m, and is then thrown to the left, i.e., given an initial push to the left. In order for the block to knock over the bottle when it is moving to the right, it must be “thrown” with an initial speed to the left v0. Ignoring the width of the block, what is the minimum value of v0? 32. B. Three objects, a disk (ICM = ½ MR ), a hoop (ICM = MR ), and a hollow ball (ICM = b MR ) all have 2 2 2 the same mass and radius. Each is subject to the same uniform tangential force that causes the object, starting from rest, to rotate with increasing angular speed about an axis through the center of mass for each object. In the case of the hollow ball the tangential force has a moment arm equal to the radius of the ball. In the space below, enter D for disk, H for hoop, and/or B for hollow ball, or same to best answer the question. 1. The object with the largest moment of inertia about the axis through the CM. 2. The object experiencing the greatest net torque. 3. The object with the greatest angular acceleration during the period the force is acting. 4. The object rotating with the smallest angular speed assuming the force has been acting for the same length of time on each object. 33. A. A uniform disk (D), hoop (H), and sphere (S), all with the same mass and radius, can freely rotate about an axis through the center of mass (CM) of each. A massless string is wrapped around each item. The string is used to apply a constant and equal tangential force to each object. See figure. For the statements below, enter D, H, S, none or the same. Assume all objects start from rest at the same instant. 1. The one with the smallest moment of inertia about the shown axis. 2. The object experiencing the largest net torque. 3. The object undergoing the smallest angular acceleration. 4. The object with the largest angular speed after an elapsed time of 5.0 s. 5. The object for which the largest amount of string has unraveled in 5.0 s. 6. The object with the smallest KErot after 5.0 s. 7. The object that undergoes the most rotations in 5.0 s. B. A spherical object is completely immersed in a liquid of density Dliq some distance above the bottom of the vessel. See figure. The upper surface is initially open to the earth’s atmosphere at sea level. Assume the liquid and object are both incompressible. For the items below, indicate whether the object sinks to the bottom (B), rises to the surface (T), or does nothing (N). 1. The vessel is brought to Salt Lake City. 2. Salt is dissolved in the liquid in the same way fresh water is turned into salt water. 3. The top 50 cm3 of the liquid is removed from the vessel. 4. The entire apparatus is transported to the surface of the moon. 5. The volume of the spherical object is increased by heating it without heating the liquid. 6. The spherical object is moved 10 cm farther down in the vessel and released. 7. A mass is placed on the top surface of the liquid in the vessel increasing the pressure at the surface. No fluid leaks. 34. A 2.20 × 103 N uniform beam is attached to an overhead beam as shown in the drawing. A 3.60 × 103 N trunk hangs from an attachment to the beam two-thirds of the way down from the upper connection of the beam to the overhead support. A cable is tied to the lower end of the beam and is also attached to the wall on the right. A. What is the tension in the cable connecting the lower end of the beam to the wall? B. What are magnitude of the vertical and horizontal components of the force the overhead beam exerts on the upper end of the beam at P? 35. A. A 12.0 kg block moves back and forth on a frictionless horizontal surface between two springs. The spring on the right has a force constant k = 825 N/m. When the block arrives at the spring on the right, it compresses that spring 0.180 m from its unstretched position. 1. What is the total mechanical energy of the block and two spring system? 2. With what speed does the block travel between the two springs while not in contact with either spring? 3. Suppose the block, after arriving at the left spring, remains in contact with that spring for a total time of 0.650 s, before separating on its way to the right spring? Using the connection between this 0.650 s and the period of oscillation between the block and the left spring, determine the spring constant of the left spring. B. A turkey baster (see figure) consists of a squeeze bulb attached to a plastic tube. When the bulb is squeezed and released, with the open end of the tube under the surface of the turkey gravy, the gravy rises in the tube to a distance h, as shown in the drawing. It can then be squirted over the turkey. Using Patm = 1.013 × 105 N/m2 for atmospheric pressure and 1.10 × 103 kg/m3 for the density of the gravy, determine the absolute pressure of the air in the bulb with the distance h = 0.160 m. Give answer to three significant digits. 36. A. The pictures below depict three glass vessels, each filled with a liquid. The liquids each have different densities, and DA > DB > DC. In vessel C an unknown block is neutrally buoyant halfway to the bottom and completely submerged. A, B, and/or C, or none are all possible answers. 1. _______ In which vessel(s) would the block sink all the way to the bottom? 2. _______ In which vessel(s) would the largest volume of the block be exposed above the surface of the liquid? 3. _______ In which vessel(s) would the buoyant forces on the block be the same? B. A swinging pendulum (A) and a mass-spring system (B) are built to have identical periods. For the statements below enter either A, B, U (unchanged) to best fit which oscillating system would have the larger period as a result of the change. 1. _______ The mass of the mass-spring system is increased. 2. _______ The mass of the swinging pendulum is increased without altering the location of its center of mass. 3. _______ The spring constant of the mass-spring system is increased. 4. _______ The length of the swinging pendulum system is increased. 5. _______ Both systems are taken to the moon and set oscillating. C. A block of mass m moves back and forth on a frictionless surface between two springs. See drawing. Assume kL > kR. For the statements below enter L for the left spring, R for the right spring, or same as the case may be. 1. _______ The spring that has the maximum compression when m is momentarily at rest. 2. _______ The spring that stores the larger elastic potential energy when maximally compressed. 3. _______ The spring that momentarily stops the block in the least time once the block arrives at the spring. 37. A uniform beam extending at right angles from a wall is used to display an advertising sign for an eatery. The beam is 2.50 m long an weighs 80.0 N. The sign, whose dimensions are 1.00 m by 0.800 m, is uniform, and weighs 200. N, hangs from the beam as shown in the drawing. A cable, attached to the wall of the eatery at a point on the beam where the inside end of the sign is attached to the beam and making an angle of 60.0° with the beam, supports this advertising structure. A. What is the magnitude of the tension in the cable supporting the beam? B. What are the magnitudes of the horizontal and vertical forces the wall exerts on the left end of the beam? 38. A. Examine the picture shown to the right. Initially, before the pump is turned on, the two masses (m1 = 1.00 kg, m2 = 2.75 kg) are held in place. the pressures above and below m1 are Patm = 1.01 × 10 N/m and 5 2 the spring is in its unstretched position. The pump is turned on and the masses are allowed to move. The mass m1 moves without friction inside a cylindrical piston of radius r = 3.85 cm. Once equilibrium is established, by what distance has the spring stretched? Take k = 2.00 × 103 N/m for the spring constant. B. A solid cylinder (radius 0.125 m and height 0.150 m) has a mass of 6.50 kg. The cylinder is floating in water. Oil (Doil = 725 kg/m ) is poured on top of the water until 3 the situation shown in the drawing results. How much of the height (in meters) of the cylinder remains in the water layer?

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## /The following graphs depict the motion of an object starting from rest and moving without friction. Describe how you would calculate the object’s acceleration, instantaneous speed, and distance at time “p” from each graph (slope, area-under-curve, etc.). 15. An object is launched at an angle of 450 from the ground of a mystery planet. The object hits the ground 20m away after a total flight time of 4.0s. Assume no air resistance. a. What are the initial vertical and horizontal velocities? b. Calculate the acceleration due to gravity. c. Draw graphs to quantitatively represent the vertical and horizontal velocities for the entire 4.0s of flight. Linear Dynamics 1. A block of mass 3 kg, initially at rest, is pulled along a frictionless, horizontal surface with a force shown as a function of time by the graph above. Calculate the acceleration and speed after 2s. Questions 2-4: Two blocks of masses M and m, with M > m, are connected by a light string. The string passes over a frictionless pulley of negligible mass so that the blocks hang vertically like Atwood’s machine. The blocks are then released from rest as shown above. 2. Draw a free-body diagram for each mass. Compare and contrast the tension on each. 3. Compare and contrast the net-force acting on each block. 4. Draw a free-body diagram for the string holding the pulley. Explain whether the force increases, decreases, or remains the same as the blocks accelerate. Questions 5-6: A ball is released from the top of a curved hill as shown above; the hill has sufficient friction so that the ball rolls as it moves down the hill. 5. What can be inferred about the ball’s linear acceleration and speed as the ball goes from the top to the bottom? (Increase, decrease, or remain the same) 6. Draw a free-body diagram for each location in the diagram to compare the weight, normal, and friction forces as it rolls down hill. Questions 7-8: Consider the above block sitting on a smooth tabletop. It is connected by a light string that passes over a frictionless and massless pulley to a pulling force of 30N downward. 7. Use Newton’s 2nd Law to determine what will happen to the net force, mass, and acceleration of the entire system if the pulling force of 30N is replaced with another block weighing 30N. 8. What will happen to the tension on each body?

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