Question: How might you determine experimentally the moment

> A small rock with mass 0.12 kg is fastened to a massless string with length 0.80 m to form a pendulum. The pendulum is swinging so as to make a maximum angle of 45° with the vertical. Air resistance is negligible. (a) What is the speed of the rock when t

> You are asked to design a spring that will give a 1160-kg satellite a speed of 2.50 m/s relative to an orbiting space shuttle. Your spring is to give the satellite a maximum acceleration of 5.00g. The spring’s mass, the recoil kinetic energy of the shutt

> A small block with mass 0.0500 kg slides in a vertical circle of radius R = 0.800 m on the inside of a circular track. There is no friction between the track and the block. At the bottom of the block’s path, the normal force the track exerts on the block

> A small block with mass 0.0400 kg slides in a vertical circle of radius R = 0.500 m on the inside of a circular track. During one of the revolutions of the block, when the block is at the bottom of its path, point A, the normal force exerted on the block

> A 0.500-kg block, attached to a spring with length 0.60 m and force constant 40.0 N>m, is at rest with the back of the block at point A on a frictionless, horizontal air table (Fig. P7.69). The mass of the spring is negligible. You move the block to t

> You are designing an amusement park ride. A cart with two riders moves horizontally with speed v = 6.00 m/s. You assume that the total mass of cart plus riders is 300 kg. The cart hits a light spring that is attached to a wall, momentarily comes to rest

> A 3.00-kg fish is attached to the lower end of a vertical spring that has negligible mass and force constant 900 N/m. The spring initially is neither stretched nor compressed. The fish is released from rest. (a) What is its speed after it has descended 0

> A basket of negligible weight hangs from a vertical spring scale of force constant 1500 N/m. (a) If you suddenly put a 3.0-kg adobe brick in the basket, find the maximum distance that the spring will stretch. (b) If, instead, you release the brick from 1

> You are an industrial engineer with a shipping company. As part of the package-handling system, a small box with mass 1.60 kg is placed against a light spring that is compressed 0.280 m. The spring has force constant k = 45.0 N/m. The spring and box are

> If a fish is attached to a vertical spring and slowly lowered to its equilibrium position, it is found to stretch the spring by an amount d. If the same fish is attached to the end of the unstretched spring and then allowed to fall from rest, through wha

> A 0.150-kg block of ice is placed against a horizontal, compressed spring mounted on a horizontal tabletop that is 1.20 m above the floor. The spring has force constant 1900 N>m and is initially compressed 0.045 m. The mass of the spring is negligible. T

> A 3.00-kg block is connected to two ideal horizontal springs having force constants k1 = 25.0 N/cm and k2 = 20.0 N/cm (Fig. P7.62). The system is initially in equilibrium on a horizontal, frictionless surface. The block is now pushed 15.0 cm to the right

> A 2.50-kg block on a horizontal floor is attached to a horizontal spring that is initially compressed 0.0300 m. The spring has force constant 840 N/m. The coefficient of kinetic friction between the floor and the block is

> A conservative force F is in the +x-direction and has magnitude F(x)= a/(x + x0)2, where a = 0.800 N#m2 and x0 = 0.200 m. (a) What is the potential-energy function U1x2 for this force? Let U(x) → 0 as x → ∞. (b) An object with mass m = 0.500 kg is relea

> A sled with rider having a combined mass of 125 kg travels over a perfectly smooth icy hill (Fig. P7.60). How far does the sled land from the foot of the cliff? Fig. P7.60: Figure P7.60 11.0 m Cliff 22.5 m/s

> A certain spring found not to obey Hooke’s law exerts a restoring force Fx(x)= -ax - bx2 if it is stretched or compressed, where a = 60.0 N/m and b = 18.0 N/m2. The mass of the spring is negligible. (a) Calculate the potential-energy function U(x) for th

> A truck with mass m has a brake failure while going down an icy mountain road of constant downward slope angle a (Fig. P7.58). Initially the truck is moving downhill at speed v0. After careening downhill a distance L with negligible friction, the truck d

> In a truck-loading station at a post office, a small 0.200-kg package is released from rest at point A on a track that is one quarter of a circle with radius 1.60 m (Fig. P7.57). The size of the package is much less than 1.60 m, so the package can be tre

> A ball is thrown upward with an initial velocity of 15 m/s at an angle of 60.0° above the horizontal. Use energy conservation to find the ball’s greatest height above the ground.

> A skier starts at the top of a very large, frictionless snowball, with a very small initial speed, and skis straight down the side (Fig. P7.55). At what point does she lose contact with the snowball and fly off at a tangent? That is, at the instant she l

> A 60.0-kg skier starts from rest at the top of a ski slope 65.0 m high. (a) If friction forces do -10.5 kJ of work on her as she descends, how fast is she going at the bottom of the slope? (b) Now moving horizontally, the skier crosses a patch of soft sn

> A 0.300-kg potato is tied to a string with length 2.50 m, and the other end of the string is tied to a rigid support. The potato is held straight out horizontally from the point of support, with the string pulled taut, and is then released. (a) What is t

> During the calibration process, the cantilever is observed to deflect by 0.10 nm when a force of 3.0 pN is applied to it. What deflection of the cantilever would correspond to a force of 6.0 pN? (a) 0.07 nm; (b) 0.14 nm; (c) 0.20 nm; (d) 0.40 nm.

> A 2.50-kg mass is pushed against a horizontal spring of force constant 25.0 N/cm on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to sto

> The fish shoots the drop of water at an insect that hovers on the water’s surface, so just before colliding with the insect, the drop is still moving at the speed it had when it left the fish’s mouth. In the collision, the drop sticks to the insect, and

> A new species of eel is found to have the same mass but one-quarter the length and twice the diameter of the American eel. How does its moment of inertia for spinning around its long axis compare to that of the American eel? The new species has (a) half

> The eel has a certain amount of rotational kinetic energy when spinning at 14 spins per second. If it swam in a straight line instead, about how fast would the eel have to swim to have the same amount of kinetic energy as when it is spinning? (a) 0.5 m/s

> The eel is observed to spin at 14 spins per second clockwise, and 10 seconds later it is observed to spin at 8 spins per second counterclockwise. What is the magnitude of the eel’s average angular acceleration during this time? (a) 6/10 rad/s2; (b) 6

> The stage moves at a constant speed while stretching the DNA. Which of the graphs in Fig. P7.84 best represents the power supplied to the stage versus time? Fig. P7.84: Figure P7.84 (a) (b) (d) Time Time Time Time Power

> Based on Fig. P7.82, how much elastic potential energy is stored in the DNA when it is stretched 50 nm? (a) 2.5 × 10-19 J; (b) 1.2 × 10-19 J; (c) 5.0 × 10-12 J; (d) 2.5 × 10-12 J. Figure P7.82:

> A segment of DNA is put in place and stretched. Figure P7.82 shows a graph of the force exerted on the DNA as a function of the displacement of the stage. Based on this graph, which statement is the best interpretation of the DNA’s beha

> A projectile has the same initial kinetic energy no matter what the angle of projection. Why doesn’t it rise to the same maximum height in each case?

> Which of the following formulas is valid if the angular acceleration of an object is not constant? Explain your reasoning in each case. (a) v = rw; (b) atan = ra; (c) w = w0 + at; (d) atan = rw2; (e) K = 1 2 Iw2.

> Can you think of a body that has the same moment of inertia for all possible axes? If so, give an example, and if not, explain why this is not possible. Can you think of a body that has the same moment of inertia for all axes passing through a certain po

> (a) For the elevator of Example 7.9 (Section 7.2), what is the speed of the elevator after it has moved downward 1.00 m from point 1 in Fig. 7.17? (b) When the elevator is 1.00 m below point 1 in Fig. 7.17, what is its acceleration?

> The food calorie, equal to 4186 J, is a measure of how much energy is released when the body metabolizes food. A certain fruit­and­cereal bar contains 140 food calories. (a) If a 65­kg hiker eats one bar, how high a mountain must he climb to “work off” t

> You are designing a flywheel to store kinetic energy. If all of the following uniform objects have the same mass and same angular velocity, which one will store the greatest amount of kinetic energy? Which will store the least? Explain. (a) A solid spher

> What is the purpose of the spin cycle of a washing machine? Explain in terms of acceleration components.

> A flywheel rotates with constant angular velocity. Does a point on its rim have a tangential acceleration? A radial acceleration? Are these accelerations constant in magnitude? In direction? In each case give your reasoning.

> What is the difference between tangential and radial acceleration for a point on a rotating body?

> Estimate your own moment of inertia about a vertical axis through the center of the top of your head when you are standing up straight with your arms outstretched. Make reasonable approximations and measure or estimate necessary quantities.

> A wheel is rotating about an axis perpendicular to the plane of the wheel and passing through the center of the wheel. The angular speed of the wheel is increasing at a constant rate. Point A is on the rim of the wheel and point B is midway between the r

> A diatomic molecule can be modeled as two point masses, m1 and m2, slightly separated (Fig. Q9.2). If the molecule is oriented along the y-axis, it has kinetic energy K when it spins about the x-axis. What will its kinetic energy (in terms of K) be if it

> You can use any angular measure—radians, degrees, or revolutions—in some of the equations in Chapter 9, but you can use only radian measure in others. Identify those for which using radians is necessary and those for which it is not, and in each case giv

> An elaborate pulley consists of four identical balls at the ends of spokes extending out from a rotating drum (Fig. Q9.18). A box is connected to a light, thin rope wound around the rim of the drum. When it is released from rest, the box acquires a speed

> Two identical balls, A and B, are each attached to very light string, and each string is wrapped around the rim of a frictionless pulley of mass M. The only difference is that the pulley for ball A is a solid disk, while the one for ball B is a hollow di

> A spring of negligible mass has force constant k = 1600 N/m. (a) How far must the spring be compressed for 3.20 J of potential energy to be stored in it? (b) You place the spring vertically with one end on the floor. You then drop a 1.20-kg book onto it

> For the equations for I given in parts (a) and (b) of Table 9.2 to be valid, must the rod have a circular cross section? Is there any restriction on the size of the cross section for these equations to apply? Explain. Table 9.2: TABLE 9.2 Moments o

> Describe how you could use part (b) of Table 9.2 to derive the result in part (d). Table 9.2: TABLE 9.2 Moments of Inertia of Various Bodies (a) Slender rod, axis through center (b) Slender rod, axis through one end (c) Rectangular plate, axis thro

> In a completely inelastic collision between two objects, where the objects stick together after the collision, is it possible for the final kinetic energy of the system to be zero? If so, give an example in which this would occur. If the final kinetic en

> (a) If the momentum of a single point object is equal to zero, must the object’s kinetic energy also be zero? (b) If the momentum of a pair of point objects is equal to zero, must the kinetic energy of those objects also be zero? (c) If the kinetic energ

> When rain falls from the sky, what happens to its momentum as it hits the ground? Is your answer also valid for Newton’s famous apple?

> When an object breaks into two pieces (explosion, radioactive decay, recoil, etc.), the lighter fragment gets more kinetic energy than the heavier one. This is a consequence of momentum conservation, but can you also explain it by using Newton’s laws of

> At the highest point in its parabolic trajectory, a shell explodes into two fragments. Is it possible for both fragments to fall straight down after the explosion? Why or why not?

> Suppose you catch a baseball and then someone invites you to catch a bowling ball with either the same momentum or the same kinetic energy as the baseball. Which would you choose? Explain.

> A very heavy SUV collides head-on with a very light compact car. Which of these statements about the collision are correct? (a) The amount of kinetic energy lost by the SUV is equal to the amount of kinetic energy gained by the compact; (b) the amount of

> A 1.20-kg piece of cheese is placed on a vertical spring of negligible mass and force constant k = 1800 N/m that is compressed 15.0 cm. When the spring is released, how high does the cheese rise from this initial position? (The cheese and the spring are

> Two objects of mass M and 5M are at rest on a horizontal, frictionless table with a compressed spring of negligible mass between them. When the spring is released, which of the following statements are true? (a) The two objects receive equal magnitudes o

> Two pieces of clay collide and stick together. During the collision, which of these statements are true? (a) Only the momentum of the clay is conserved; (b) only the mechanical energy of the clay is conserved; (c) both the momentum and the mechanical ene

> An apple falls from a tree and feels no air resistance. As it is falling, which of these statements about it are true? (a) Only its momentum is conserved; (b) only its mechanical energy is conserved; (c) both its momentum and its mechanical energy are co

> The net force on a particle of mass m has the potential­ energy function graphed in Fig. 7.24a. If the total energy is E1, graph the speed v of the particle versus its position x. At what value of x is the speed greatest? Sketch v versus x if

> A particle is in neutral equilibrium if the net force on it is zero and remains zero if the particle is displaced slightly in any direction. Sketch the potential ­ energy function near a point of neutral equilibrium for the case of one ­ dimensional moti

> Explain why the points x = A and x = -A in Fig. 7.23b are called turning points. How are the values of E and U related at a turning point? Fig. 7.23b: (b) On the graph, the limits of motion are the points where the U curve intersects the horizontal

> An egg is released from rest from the roof of a building and falls to the ground. As the egg falls, what happens to the momentum of the system of the egg plus the earth?

> A tennis player hits a tennis ball with a racket. Consider the system made up of the ball and the racket. Is the total momentum of the system the same just before and just after the hit? Is the total momentum just after the hit the same as 2 s later, whe

> Figure 7.22a shows the potential ­ energy function for the force Fx = -kx. Sketch the potential ­ energy function for the force Fx = +kx. For this force, is x = 0 a point of equilibrium? Is this equilibrium stable or unstable? Expla

> Since only changes in potential energy are important in any problem, a student decides to let the elastic potential energy of a spring be zero when the spring is stretched a distance x1. The student decides, therefore, to let U = 1 2 k(x - x1)2. Is this

> A spring of negligible mass has force constant k = 800 N/m. (a) How far must the spring be compressed for 1.20 J of potential energy to be stored in it? (b) You place the spring vertically with one end on the floor. You then lay a 1.60-kg book on top of

> In each of Examples 8.10, 8.11, and 8.12 (Section 8.4), verify that the relative velocity vector of the two bodies has the same magnitude before and after the collision. In each case, what happens to the direction of the relative velocity vector? Exampl

> A box slides down a ramp and work is done on the box by the forces of gravity and friction. Can the work of each of these forces be expressed in terms of the change in a potential ­ energy function? For each force explain why or why not.

> In Fig. 8.23b, the kinetic energy of the Ping-Pong ball is larger after its interaction with the bowling ball than before. From where does the extra energy come? Describe the event in terms of conservation of energy. Fig. 8.23b: (b) Moving bowling

> Two objects with different masses are launched vertically into the air by placing them on identical compressed springs and then releasing the springs. The two springs are compressed by the same amount before launching. Ignore air resistance and the masse

> A 1.0­kg stone and a 10.0­kg stone are released from rest at the same height above the ground. Ignore air resistance. Which of these statements about the stones are true? Justify each answer. (a) Both have the same initial gravitational potential energy.

> (a) A block of wood is pushed against a spring, which is compressed 0.080 m. Does the force on the block exerted by the spring do positive or negative work? Does the potential energy stored in the spring increase or decrease? (b) A block of wood is place

> (a) A book is lifted upward a vertical distance of 0.800 m. During this displacement, does the gravitational force acting on the book do positive work or negative work? Does the gravitational potential energy of the book increase or decrease? (b) A can o

> In Example 8.7 (Section 8.3), where the two gliders of Fig. 8.18 stick together after the collision, the collision is inelastic because K2 < K1 . In Example 8.5 (Section 8.2), is the collision inelastic? Explain. Example 8.7: We repeat the collision de

> A woman holding a large rock stands on a frictionless, horizontal sheet of ice. She throws the rock with speed v0 at an angle a above the horizontal. Consider the system consisting of the woman plus the rock. Is the momentum of the system conserved? Why

> Is it possible for a friction force to increase the mechanical energy of a system? If so, give examples.

> A slingshot will shoot a 10-g pebble 22.0 m straight up. (a) How much potential energy is stored in the slingshot’s rubber band? (b) With the same potential energy stored in the rubber band, how high can the slingshot shoot a 25-g pebble? (c) What physic

> A physics teacher had a bowling ball suspended from a very long rope attached to the high ceiling of a large lecture hall. To illustrate his faith in conservation of energy, he would back up to one side of the stage, pull the ball far to one side until t

> An egg is released from rest from the roof of a building and falls to the ground. Its fall is observed by a student on the roof of the building, who uses coordinates with origin at the roof, and by a student on the ground, who uses coordinates with origi

> An object is released from rest at the top of a ramp. If the ramp is frictionless, does the object’s speed at the bottom of the ramp depend on the shape of the ramp or just on its height? Explain. What if the ramp is not frictionless?

> A proton with mass m moves in one dimension. The potential-energy function is U(x)=(a/x2)-(b/x), where a and b are positive constants. The proton is released from rest at x0 = a/b. (a) Show that U(x) can be written as Graph U(x). Calculate U(x0) and th

> Use the methods of Challenge Problem 8.104 to calculate the x- and y-coordinates of the center of mass of a semicircular metal plate with uniform density r and thickness t. Let the radius of the plate be a. The mass of the plate is thus M = 1/2 rpa2t. Us

> In Section 8.5 we calculated the center of mass by considering objects composed of a finite number of point masses or objects that, by symmetry, could be represented by a finite number of point masses. For a solid object whose mass distribution does not

> In a rocket- propulsion problem the mass is variable. Another such problem is a raindrop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raind

> On a compact disc (CD), music is coded in a pattern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a CD player, the track is scanned at a constant linear speed of v = 1.25m/s. Because the radius

> Calculate the moment of inertia of a uniform solid cone about an axis through its center (Fig. P9.90). The cone has mass M and altitude h. The radius of its circular base is R. Fig. P9.90: Figure P9.90 h Axis

> A bicycle wheel has an initial angular velocity of 1.50 rad/s. (a) If its angular acceleration is constant and equal to 0.200 rad/s2, what is its angular velocity at t = 2.50 s? (b) Through what angle has the wheel turned between t = 0 and t = 2.50 s?

> A spring stores potential energy U0 when it is compressed a distance x0 from its uncompressed length. (a) In terms of U0, how much energy does the spring store when it is compressed (i) twice as much and (ii) half as much? (b) In terms of x0, how much mu

> A wheel is rotating about an axis that is in the z-direction. The angular velocity

> The angle u through which a disk drive turns is given by

> At t = 0 the current to a dc electric motor is reversed, resulting in an angular displacement of the motor shaft given by

> A slender rod with length L has a mass per unit length that varies with distance from the left end, where x = 0, according to dm/dx = gx, where g has units of kg/m2. (a) Calculate the total mass of the rod in terms of g and L. (b) Use Eq. (9.20) to calcu

> Use Eq. (9.20) to calculate the moment of inertia of a slender, uniform rod with mass M and length L about an axis at one end, perpendicular to the rod.

> Use Eq. (9.20) to calculate the moment of inertia of a uniform, solid disk with mass M and radius R for an axis perpendicular to the plane of the disk and passing through its center.

> A thin uniform rod of mass M and length L is bent at its center so that the two segments are now perpendicular to each other. Find its moment of inertia about an axis perpendicular to its plane and passing through (a) the point where the two segments mee

> A thin, rectangular sheet of metal has mass M and sides of length a and b. Use the parallel-axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet and that passes through one corner of the

> (a) For the thin rectangular plate shown in part (d) of Table 9.2, find the moment of inertia about an axis that lies in the plane of the plate, passes through the center of the plate, and is parallel to the axis shown. (b) Find the moment of inertia of

> A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to