Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 9

Linear Momentum and Collisions - all with Video Answers

Educators

Chapter Questions

Problem 1

A particle of mass $m$ moves with momentum of mag- nitude $p$ (a) Show that the kinetic energy of the particle is $K=p^{2} / 2 m .$ (b) Express the magnitude of the particle's momentum in terms of its kinetic energy and mass.

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Problem 2

An object has a kinetic energy of 275 $\mathrm{J}$ and a momentum of magnitude 25.0 $\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ . Find the speed and mass of the object.

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Problem 3

At one instant, a $17.5-\mathrm{kg}$ sled is moving over a horizontal surface of snow at 3.50 $\mathrm{m} / \mathrm{s}$ . After 8.75 s has elapsed, the sled stops. Use a momentum approach to find the average friction force acting on the sled while it was moving.

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Problem 4

A baseball approaches home plate at a speed of 45.0 $\mathrm{m} / \mathrm{s}$ , moving horizontally just before being hit by a bat. The batter hits a pop-up such that after hitting the bat, the baseball is moving at 55.0 $\mathrm{m} / \mathrm{s}$ straight up. The ball has a mass of 145 $\mathrm{g}$ and is in contact with the bat for 2.00 $\mathrm{ms}$ . What is the average vector force the ball exerts on the bat during their interaction?

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Problem 5

A 65.0-kg boy and his 40.0-kg sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backward with velocity 2.90 m/s toward the west. Ignore friction. (a) Describe the subsequent motion of the girl. (b) How much potential energy in the girl’s body is converted into mechanical energy of the boy–girl system? (c) Is the momentum of the boy–girl system conserved in the pushing-apart process? If so, explain how that is possible considering (d) there are large forces
acting and (e) there is no motion beforehand and plenty of motion afterward.

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Problem 6

A 45.0 -kg girl is standing on a $150-\mathrm{kg}$ plank. Both are originally at rest on a frozen lake that constitutes a frictionless, flat surface. The girl begins to walk along the plank at a constant velocity of 1.50 i $\mathrm{m} / \mathrm{s}$ relative to the plank. (a) What is the velocity of the plank relative to the ice surface? (b) What is the girl's velocity relative to the ice surface?

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Problem 7

A girl of mass $m_{g}$ is standing on a plank of mass $m_{p}$ Both are originally at rest on a frozen lake that constitutes a frictionless, flat surface. The girl begins to walk along the plank at a constant velocity $v_{g p}$ to the right relative to the plank. (The subscript gp denotes the girl relative to plank. (a) What is the velocity $v_{p i}$ of the plank relative to the surface of the ice? (b) What is the girl's velocity $v_{g i}$ relative to the ice surface?

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Problem 8

When you jump straight up as high as you can, what is the order of magnitude of the maximum recoil speed that you give to the Earth? Model the Earth as a perfectly solid object. In your solution, state the physical quantities you take as data and the values you measure or estimate for them.

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Problem 9

Two blocks of masses $m$ and 3$m$ are placed on a frictionless, horizontal surface. A light spring is attached to the more massive block, and the blocks are pushed together with the spring between them (Fig. P9.9). A cord initially holding the blocks together is burned; after that happens, the block of mass 3 $\mathrm{m}$ moves to the right with a speed of 2.00 $\mathrm{m} / \mathrm{s}$ (a) What is the velocity of the block of mass $\mathrm{m}$ ? (b) Find the system's original elastic potential energy, taking $m=0.350 \mathrm{kg}$ . (c) Is the original energy in the explain your answer cord? (d) Explain your answer to part (c). (e) Is the momentum of the system conserved in the how that is possible considering
(f) there are large forces acting and (g) there is no motion beforehand and plenty of motion afterward?

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Problem 10

A man claims that he can hold onto a 12.0 -kg child in a head-on collision as long as he has his seat belt on. Consider this man in a collision in which he is in one of two identical cars each traveling toward the other at 60.0 $\mathrm{mi} / \mathrm{h}$ relative to the ground. The car in which he rides is brought
to rest in 0.10 s. (a) Find the magnitude of the average force needed to hold onto the child. (b) Based on your result to part (a), is the man's claim valid? (c) What does the answer to this problem say about laws requiring the use of proper safety devices such as seat belts and special toddler seats?

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Problem 11

An estimated force-time curve for a baseball struck by a bat is shown in Figure P9. 11. From this curve, determine (a) the magnitude of the impulse delivered to the ball and (b) the average force exerted on the ball.

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Problem 12

Review. After a 0.300-kg rubber ball is dropped from a height of 1.75 m, it bounces off a concrete floor and
rebounds to a height of 1.50 m. (a) Determine the magnitude and direction of the impulse delivered to the ball by the floor. (b) Estimate the time the ball is in contact with the floor and use this estimate to calculate the average force the floor exerts on the ball.

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Problem 13

A glider of mass $m$ is free to slide along a horizontal air track. It is pushed against a launcher at one
end of the track. Model the launcher as a light spring of force constant $k$ compressed by a distance $x$ . The glider is released from rest. (a) Show that the glider attains a speed of $v=x(k / m)^{1 / 2}$ . (b) Show that the magnitude of the impulse imparted to the glider is given by the expression $I=x(k m)^{1 / 2} .$ (c) Is more work done on a cart with a large or a small mass?

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Problem 14

A tennis player receives a shot with the ball $(0.0600 \mathrm{kg})$ traveling horizontally at 50.0 $\mathrm{m} / \mathrm{s}$ and returns the shot with the ball traveling horizontally at 40.0 $\mathrm{m} / \mathrm{s}$ in the opposite direction. (a) What is the impulse delivered to the ball by the tennis racquet? (b) What work does the racquet do on the ball?

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Problem 15

The magnitude of the net force exerted in the x direction on a 2.50-kg particle varies in time as shown in Figure P9.15. Find (a) the impulse of the force over the 5.00-s time interval, (b) the final velocity the particle attains if it is originally at rest, (c) its final velocity if its original velocity is $-2.00 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}$ , and $(\mathrm{d})$ the average force exerted on the particle for the time interval between 0 and 5.00 $\mathrm{s}$ .

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Problem 16

Review. A force platform is a tool used to analyze the performance of athletes by measuring the vertical force the athlete exerts on the ground as a function of time. Starting from rest, a $65.0-\mathrm{kg}$ athlete jumps down onto the platform from a height of 0.600 $\mathrm{m}$ . While she is in contact with the platform during the time interval $0 < t < 0.800 \mathrm{s}$ , the force she exerts on it is described by the function
$$
F=9200 t-11500 t^{2}
$$
where $F$ is in newtons and $t$ is in seconds. (a) What impulse did the athlete receive from the platform? (b) With what speed did she reach the platform? (c) With what speed did she leave it? (d) To what height did she jump upon leaving the platform?

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Problem 17

Water falls without splashing at a rate of 0.250 $\mathrm{L} / \mathrm{s}$ from a height of 2.60 $\mathrm{m}$ into a 0.750 -kg bucket on a scale. If the bucket is originally empty, what does the scale read in newtons 3.00 s after water starts to accumulate in it?

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Problem 18

A 1200 -kg car traveling initially at $v_{\mathrm{C} i}=25.0 \mathrm{m} / \mathrm{s}$ in an easterly direction crashes into the back of a $9000-\mathrm{kg}$ truck moving in the same direction at $v_{\mathrm{Ti}}=20.0 \mathrm{m} / \mathrm{s}$ (Fig. P9.18). The velocity of the car immediately after the collision is $v_{\mathrm{C} f}=18.0 \mathrm{m} / \mathrm{s}$ to the east. ( a) What is the velocity of the truck immediately after the collision? (b) What is the change in mechanical energy of the car-truck system in the collision? (c) Account for this change in mechanical energy.

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Problem 19

A 10.0 -g bullet is fired into a stationary block of wood having mass $m=5.00 \mathrm{kg}$ . The bullet imbeds into the block. The speed of the bullet-plus-wood combination immediately after the collision is 0.600 $\mathrm{m} / \mathrm{s}$ . What was the original speed of the bullet?

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Problem 20

A car of mass $m$ moving at a speed $v_{1}$ collides and couples with the back of a truck of mass 2 $\mathrm{m}$ moving initially in the same direction as the car at a lower speed $v_{2} .$ (a) What is the speed $v_{f}$ of the two vehicles immediately after the collision? (b) What is the change in kinetic energy of the car-truck system in the collision?

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Problem 21

A neutron in a nuclear reactor makes an elastic, head- on collision with the nucleus of a carbon atom initially at rest. (a) What fraction of the neutron’s kinetic energy is transferred to the carbon nucleus? (b) The initial kinetic energy of the neutron is $1.60 \times 10^{-13} \mathrm{J}$ . Find its final kinetic energy and the kinetic energy of the carbon nucleus after the collision. (The mass of the carbon nucleus is nearly
12.0 times the mass of the neutron.)

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Problem 22

A tennis ball of mass $m_{t}$ is held just above a basketball of mass $m_{b},$ as shown in Figure P9.22. With their centers vertically aligned, both are released from rest at the same moment so that the bottom of the basketball falls freely through a height $h$ and strikes the floor. Assume an elastic collision with the ground instantaneously reverses the velocity of the basketball while the tennis ball is still moving down because the balls have separated a bit while falling. Next, the two balls meet in an elastic collision. (a) To what height does the tennis ball rebound? (b) How do you account for the height in (a) being larger than $h$ ? Does that seem like a violation of conservation of energy?

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Problem 23

A 12.0-g wad of sticky clay is hurled horizontally at a 100 -g wooden block initially at rest on a horizontal surface. The clay sticks to the block. After impact, the block slides 7.50 $\mathrm{m}$ before coming to rest. If the coefficient of friction between the block and the surface is $0.650,$ what was the speed of the clay immediately before impact?

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Problem 24

A wad of sticky clay of mass $m$ is hurled horizontally at a wooden block of mass $M$ initially at rest on a horizontal surface. The clay sticks to the block. After impact, the block slides a distance $d$ before coming to rest. If the coefficient of friction between the block and the surface is $\mu$ what was the speed of the clay immediately before impact?

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Problem 25

(a) Three carts of masses $m_{1}=4.00 \mathrm{kg}, m_{2}=10.0 \mathrm{kg}$ and $m_{3}=3.00 \mathrm{kg}$ move on a frictionless, horizontal track with speeds of $v_{1}=5.00 \mathrm{m} / \mathrm{s}$ to the right, $v_{2}=3.00 \mathrm{m} / \mathrm{s}$ to the right, and $v_{3}=4.00 \mathrm{m} / \mathrm{s}$ to the left as shown in Figure P9.25. Velcro couplers make the carts stick together after colliding. Find the final velocity of the train of three carts. (b) What If? Does your answer in part (a) require that all the carts collide and stick together at the same moment? What if they collide in a different order?

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Problem 26

As shown in Figure $\mathrm{P} 9.26$ a bullet of mass $m$ and speed $v$ passes completely through a pendulum bob of mass $M$ . The bullet emerges with a speed of $v / 2 .$ The pendulum bob is suspended by a stiff rod $(n o t \text { a string) of length }$ $\ell$ and negligible mass. What is the minimum value of $v$ such that the pendulum bob will barely swing through a complete vertical circle?

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Problem 27

Two blocks are free to slide along the frictionless, wooden track shown in Figure $\mathrm{P} 9.27$ . The block of mass $m_{1}=$ 5.00 $\mathrm{kg}$ is released from the position shown, at height $h=$ 5.00 $\mathrm{m}$ above the flat part of the track. Protruding from its front end is the north pole of a strong magnet, which repels the north pole of an identical magnet embedded in the back end of the block of mass $m_{2}=10.0 \mathrm{kg}$ , initially at rest. The two blocks never touch. Calculate the maximum
height to which $m_{1}$ rises after the elastic collision.

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Problem 28

Two automobiles of equal mass approach an intersection. One vehicle is traveling with speed 13.0 $\mathrm{m} / \mathrm{s}$ toward the east, and the other is traveling north with speed $v_{2 i} \cdot$ Neither driver sees the other. The vehicles collide in the intersection and stick together, leaving parallel skid marks at an angle of $55.0^{\circ}$ north of east. The speed limit for both roads is 35 $\mathrm{mi} / \mathrm{h}$ , and the driver of the northward-moving vehicle claims he was within the speed limit when the collision occurred. Is he telling the truth? Explain your reasoning.

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Problem 29

An object of mass 3.00 $\mathrm{kg}$ , moving with an initial velocity of 5.00 i $\mathrm{m} / \mathrm{s}$ , collides with and sticks to an object of mass 2.00 $\mathrm{kg}$ with an initial velocity of $-3.00 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}$ . Find the final velocity of the composite object.

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Problem 30

Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed of 5.00 m/s. After the collision, the orange disk moves along a direction that makes an angle of 37.08 with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk.

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Problem 31

Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed $v_{i}$ . After the collision, the orange disk moves along a direction that makes an angle $\theta$ with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk.

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Problem 32

A 90.0-kg fullback running east with a speed of 5.00 m/s is tackled by a 95.0-kg opponent running north
with a speed of 3.00 m/s. (a) Explain why the successful tackle constitutes a perfectly inelastic collision. (b) Calculate the velocity of the players immediately after the tackle. (c) Determine the mechanical energy that disappears as a result of the collision. Account for the missing energy.

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Problem 33

A billiard ball moving at 5.00 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 4.33 m/s at an angle of 30.08 with respect to the original line of motion. Assuming an elastic collision (and ignoring friction and rotational motion), find the struck ball’s velocity after the collision.

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Problem 34

The mass of the blue puck in Figure P9.34 is 20.0% greater than the mass of the green puck. Before colliding, the pucks approach each other with momenta of equal magnitudes and opposite directions, and the green puck has an initial speed of 10.0 m/s. Find the speeds the pucks have after the collision if half the kinetic energy of the system becomes internal energy during the collision.

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Problem 35

An unstable atomic nucleus of mass $17.0 \times 10^{-27} \mathrm{kg}$ initially at rest disintegrates into three particles. One of the particles, of mass $5.00 \times 10^{-27} \mathrm{kg}$ , moves in the $y$ direction with a speed of $6.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Another particle, of mass
$8.40 \times 10^{-27} \mathrm{kg}$ , moves in the $x$ direction with a speed of $4.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Find $(\mathrm{a})$ the velocity of the third particle and $(\mathrm{b})$ the total kinetic energy increase in the process.

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Problem 36

The mass of the Earth is $5.97 \times 10^{24} \mathrm{kg}$ , and the mass of the Moon is $7.35 \times 10^{22} \mathrm{kg}$ . The distance of separation, measured between their centers, is $3.84 \times 10^{8} \mathrm{m} .$ Locate the center of mass of the Earth-Moon system as measured from the center of the Earth.

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Problem 37

Four objects are situated along the $y$ axis as follows: a $2.00-\mathrm{kg}$ object is at $+3.00 \mathrm{m},$ a $3.00-\mathrm{kg}$ object is at $+2.50 \mathrm{m},$ a 2.50 -kg object is at the origin, and a $4.00-\mathrm{kg}$ object is at $-0.500 \mathrm{m} .$ Where is the center of mass of these objects?

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Problem 38

A uniform piece of sheet metal is shaped as shown in Figure P9.38. Compute the $x$ and $y$ coordinates of the center of mass of the piece.

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Problem 39

Explorers in the jungle find an ancient monument in the shape of a large isosceles triangle as shown in Figure P9.39. The monument is made from tens of thousands of small stone blocks of density $3800 \mathrm{kg} / \mathrm{m}^{3} .$ The monument is 15.7 $\mathrm{m}$ high and 64.8 $\mathrm{m}$
wide at its base and is everywhere 3.60 $\mathrm{m}$ thick from front to back. Before the monument was built many years ago, all the stone blocks lay on the ground. How much work did laborers do on the blocks to put them in position while building the entire monument? Note: The gravitational potential energy of an object-Earth system is given by $U_{g}=M g y_{\mathrm{CM}},$ where $M$ is the total mass of the object and $y_{\mathrm{CM}}$ is the elevation of its center of mass above the chosen reference level.

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Problem 40

A rod of length 30.0 $\mathrm{cm}$ has linear density (mass per length) given by
$$
\lambda=50.0+20.0 x
$$
where $x$ is the distance from one end, measured in meters, and $\lambda$ is in grams/meter. (a) What is the mass of the rod? (b) How far from the $x=0$ end is its center of mass?

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Problem 41

A 2.00 -kg particle has a velocity $(2.00 \mathrm{i}-3.00 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ , and a 3.00 -kg particle has a velocity $(1.00 \hat{\mathrm{i}}+6.00 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ . Find (a) the velocity of the center of mass and $(\mathrm{b})$ the total momentum of the system.

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Problem 42

The vector position of a 3.50 -g particle moving in the $x y$ plane varies in time according to $\overrightarrow{\mathbf{r}}_{1}=(3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}) t+2 \hat{\mathbf{j}} t^{2},$ where $t$ is in seconds and $\overrightarrow{\mathbf{r}}$ is in centimeters. At the same time, the vector position of a $5.50-\mathrm{g}$ particle varies as $\overrightarrow{\mathbf{r}}_{2}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{i}} t^{2}-6 \hat{\mathbf{j}} t .$ At $t=2.50$ s, determine (a) the vector position of the center of mass, (b) the linear momentum of the system, (c) the velocity of the center of mass, (d) the acceleration of the center of mass, and (e) the net force exerted on the two-particle system.

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Problem 43

Romeo (77.0 kg) entertains Juliet (55.0 kg) by playing his guitar from the rear of their boat at rest in still water, 2.70 m away from Juliet, who is in the front of the boat. After the serenade, Juliet carefully moves to the rear of the boat (away from shore) to plant a kiss on Romeo’s cheek. How far does the 80.0-kg boat move toward the shore it is facing?

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Problem 44

A ball of mass 0.200 $\mathrm{kg}$ with a velocity of 1.50 $\mathrm{i} \mathrm{m} / \mathrm{s}$ meets a ball of mass 0.300 $\mathrm{kg}$ with a velocity of $-0.400 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}$ in a head-on, elastic collision. (a) Find their velocities after the collision. (b) Find the velocity of their center of mass before and after the collision.

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Problem 45

For a technology project, a student has built a vehicle, of total mass 6.00 kg, that moves itself. As shown in Figure P9.45, it runs on four light wheels. A reel is attached to one of the axles, and a cord originally wound on the reel goes up over a pulley attached to the vehicle to support an elevated load. After the vehicle is released from rest, the load descends very slowly, unwinding the cord to turn
the axle and make the vehicle move forward (to the left in Fig. P9.45). Friction is negligible in the pulley and axle bearings. The wheels do not slip on the floor. The reel has been constructed with a conical shape so that the load descends at a constant low speed while the vehicle moves horizontally across the floor with constant acceleration, reaching a final velocity of 3.00i^m/s. (a) Does the floor impart impulse to the vehicle? If so, how much? (b) Does the floor do work on the vehicle? If so, how much? (c) Does it make sense to say that the final momentum of the vehicle came from the floor? If not, where did it come from? (d) Does it make sense to say that the final kinetic energy of the vehicle came from the floor? If not, where did it come from? (e) Can we say that one particular force causes the forward acceleration of the vehicle? What does cause it?

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Problem 46

Figure $\mathrm{P} 9.46 \mathrm{a}$ shows an overhead view of the initial configuration of two pucks of mass $m$ on frictionless ice. The pucks are tied together with a string of length $\ell$ and negligible mass. At time $t=0,$ a constant force of magnitude $F$ begins to pull to the right on the center point of the string. At time $t,$ the moving pucks strike each other and stick together. At this time, the force has moved through a distance $d,$ and the pucks have attained a speed $v(\text { Fig. } \mathrm{P} 9.46 \mathrm{b}) .$ (a) What is $v$ in terms of $F, d, \ell,$ and $m ?$ (b) How much of the energy transferred into the system by work done by the force has been transformed to internal energy?

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Problem 47

A particle is suspended from a post on top of a cart by a light string of length L as shown in Figure P9.47a. The cart and particle are initially moving to the right at constant speed $v_{i},$ with the string vertical. The cart suddenly comes to rest when it runs into and sticks to a bumper as shown in Figure $\mathrm{P} 9.47 \mathrm{b}$ . The suspended particle swings through an angle $\theta$ (a) Show that the original speed of the cart can be computed from $v_{i}=\sqrt{2 g L(1-\cos \theta)}$ . (b) If the bumper is still exerting a horizontal force on the cart when the hanging particle is at its maximum angle forward from the vertical, at what moment does the bumper stop exerting a horizontal force?

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Problem 48

A 60.0 -kg person bends his knees and then jumps straight up. After his feet leave the floor, his motion is unaffected by air resistance and his center of mass rises by a maximum of 15.0 cm. Model the floor as completely solid and motionless. (a) Does the floor impart impulse to the person? (b) Does the floor do work on the person? (c) With what momentum does the person leave the floor? (d) Does it make sense to say that this momentum came from the floor? Explain. (e) With what kinetic energy does the person leave the floor? (f) Does it make sense to say that this energy came from the floor? Explain.

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Problem 49

A model rocket engine has an average thrust of 5.26 N. It has an initial mass of 25.5 g, which includes fuel mass of 12.7 g. The duration of its burn is 1.90 s. (a) What is the average exhaust speed of the engine? (b) This engine is placed in a rocket body of mass 53.5 g. What is the final velocity of the rocket if it were to be fired from rest in outer space by an astronaut on a spacewalk? Assume the fuel burns at a constant rate.

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Problem 50

Review. The first stage of a Saturn $\mathrm{V}$ space vehicle consumed fuel and oxidizer at the rate of $1.50 \times 10^{4} \mathrm{kg} / \mathrm{s}$ with an exhaust speed of $2.60 \times 10^{3} \mathrm{m} / \mathrm{s}$ . (a) Calculate the thrust produced by this engine. (b) Find the acceleration
the vehicle had just as it lifted off the launch pad on the Earth, taking the vehicle's initial mass as $3.00 \times 10^{6} \mathrm{kg}$ .

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Problem 51

A rocket for use in deep space is to be capable of boosting a total load (payload plus rocket frame and engine) of 3.00 metric tons to a speed of 10 000 m/s. (a) It has an engine and fuel designed to produce an exhaust speed of 2 000 m/s. How much fuel plus oxidizer is required? (b) If a different fuel and engine design could give an exhaust speed of 5 000 m/s, what amount of fuel and oxidizer would be required for the same task? (c) Noting that the exhaust speed in part (b) is 2.50 times higher than that in part (a), explain why the required fuel mass is not simply smaller by a factor of 2.50.

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Problem 52

A rocket has total mass $M_{i}=360 \mathrm{kg}$ , including $M_{f}=330 \mathrm{kg}$ of fuel and oxidizer. In interstellar space, it starts from rest at the position $x=0$ , turns on its engine at time $t=0$ , and puts out exhaust with relative speed $v_{e}=1500 \mathrm{m} / \mathrm{s}$ at the constant rate $k=2.50 \mathrm{kg} / \mathrm{s}$ . The fuel will last for a burn time of $T_{b}=M_{f} / k=330 \mathrm{kg} /(2.5 \mathrm{kg} / \mathrm{s})=132 \mathrm{s}$ . (a) Show that during the burn the velocity of the rocket as a function of time is given by
$$
v(t)=-v_{e} \ln \left(1-\frac{k t}{M_{i}}\right)
$$
(b) Make a graph of the velocity of the rocket as a function of time for times running from 0 to 132 s. (c) Show that the acceleration of the rocket is
$$
a(t)=\frac{k v_{e}}{M_{i}-k t}
$$
(d) Graph the acceleration as a function of time. (e) Show that the position of the rocket is
$$
x(t)=v_{e}\left(\frac{M_{i}}{k}-t\right) \ln \left(1-\frac{k t}{M_{i}}\right)+v_{e} t
$$
(f) Graph the position during the burn as a function of time.

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Problem 53

A ball of mass $m$ is thrown straight up into the air with an initial speed $v_{i}$ . Find the momentum of the ball (a) at its maximum height and (b) halfway to its maximum height.

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Problem 54

An amateur skater of mass M is trapped in the middle of an ice rink and is unable to return to the side where there is no ice. Every motion she makes causes her to slip on the ice and remain in the same spot. She decides to try to return to safety by throwing her gloves of mass $m$ in the direction opposite the safe side. (a) She throws the gloves as hard as she can, and they leave her hand with a horizontal velocity $\overrightarrow{\mathbf{v}}_{\text { gloves. }}$ Explain whether or not she moves. If she does move, calculate her velocity $\overrightarrow{\mathbf{v}}_{\text { git }}$ relative to the Earth after she throws the gloves. (b) Discuss her motion from the point of view of the forces acting on her.

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Problem 55

A 3.00 -kg steel ball strikes a wall with a speed of 10.0 $\mathrm{m} / \mathrm{s}$ at an angle of
10.0 $\mathrm{m} / \mathrm{s}$ at an angle of $\theta=60.0^{\circ}$ with the surface. It bounces off with the same speed and angle in contact with the wall average force exerted by the wall on the ball?

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Numerade Educator

Problem 55

A 60.0-kg person running at an initial speed of $4.00 \mathrm{~m} / \mathrm{s}$ jumps onto a $120-\mathrm{kg}$ cart initially at rest(Figure $\mathrm{P} 9.55$ ). The person slides on the cart's top surface and finally comes to rest relative to the cart. The coefficient of kinetic friction between the person and the cart is 0.400 . Friction between the cart and ground can be neglected.
(a) Find the final velocity of the person and cart relative to the ground.
(b) Find the friction force acting on the person while he is sliding across the top surface of the cart
(c) How long does the friction force act on the person?
(d) Find the change in momentum of the person and the change in momentum of the cart.
(e) Determine the displacement of the person relative to the ground while he is sliding on the cart.
(f) Determine the displacement of the cart relative to the ground while the person is sliding
(g) Find the change in

Ummatul Choudary

Ummatul Choudary

Numerade Educator

Problem 56

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
A 1.25-kg wooden block rests on a table over a large hole as in Figure $P 9.57$ on page $274 .$ A 5.00 -g bullet with an initial velocity $v_{i}$ is fired upward into the bottom of the
block and remains in the block after the collision. The block and bullet rise to a maximum height of 22.0 cm. (a) Describe how you would find the initial velocity of the bullet using ideas you have learned in this chapter. (b) Calculate the initial velocity of the bullet from the information provided.

Robert Daine

Numerade Educator

Problem 58

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
A wooden block of mass $M$ rests on a table over a large hole as in Figure 9.57 . A bullet of mass $m$ with an initial velocity of $v_{i}$ is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of $h$ . (a) Describe how you would find the initial velocity of the bullet using ideas you have learned in this chapter. (b) Find an expression for the initial velocity of the bullet.

Robert Daine

Numerade Educator

Problem 59

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
Two gliders are set in motion on a horizontal air track. A spring of force constant k is attached to the back end of the second glider. As shown in Figure P9.59, the first glider, of mass $m_{1},$ moves to the right with speed $v_{1},$ and the second glider, of mass $m_{2},$ moves more slowly to the right with speed $v_{2} .$ When $m_{1}$ collides with the spring attached to $m_{2},$ the spring compresses by a distance $x_{\max },$ and the gliders then move apart again. In terms of $v_{1}, v_{2}, m_{1}, m_{2},$ and $k,$ find (a) the speed $v$ at maximum compression, (b) the maximum compression $x_{\text { max }},$ and $(c)$ the velocity of each glider after $m_{1}$ has lost contact with the spring.

Robert Daine

Numerade Educator

Problem 60

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
Pursued by ferocious wolves, you are in a sleigh with no horses, gliding without friction across an ice-covered lake. You take an action described by the equations $\begin{aligned}(270 \mathrm{kg})(7.50 \mathrm{m} / \mathrm{s}) \hat{\mathrm{i}} &=(15.0 \mathrm{kg})\left(-v_{1 f} \hat{\mathrm{i}}\right)+(255 \mathrm{kg})\left(v_{2 f} \hat{\mathrm{i}}\right) \\ v_{1 f}+v_{2 f} &=8.00 \mathrm{m} / \mathrm{s} \end{aligned}$
(a) Complete the statement of the problem, giving the data and identifying the unknowns. (b) Find the values of $v_{1 f}$ and $v_{2 f}$ . (c) Find the amount of energy that has been transformed from potential energy stored in your body to kinetic energy of the system.

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 61

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
Two blocks of masses $m_{1}=2.00 \mathrm{kg}$ and $m_{2}=4.00 \mathrm{kg}$ are released from rest at a height of $h=5.00 \mathrm{m}$ on a frictionless track as shown in Figure $\mathrm{P} 9.61 .$ When they meet on the level portion of the track, they undergo a head-on, elastic collision. Determine the maximum heights to which $m_{1}$ and $m_{2}$ rise on the curved portion of the track after the collision.

Robert Daine

Numerade Educator

Problem 62

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
Why is the following situation impossible? An astronaut, together with the equipment he carries, has a mass of 150 $\mathrm{kg}$ . He is taking a space walk outside his spacecraft, which is drifting through space with a constant velocity. The astronaut accidentally pushes against the spacecraft and begins moving away at 20.0 $\mathrm{m} / \mathrm{s}$ , relative to the spacecraft, without a tether. To return, he takes equipment off his space suit and throws it in the direction away from the spacecraft. Because of his bulky space suit, he can throw equipment at a maximum speed of 5.00 $\mathrm{m} / \mathrm{s}$ relative to himself. After throwing enough equipment, he starts moving back to the spacecraft and is able to grab onto it and climb inside.

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 63

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
A $0.400-\mathrm{kg}$ blue bead slides on a frictionless, curved wire, starting from rest at point $\mathbb{A}$ in Figure $\mathrm{P} 9.63,$ where $h=$ $1.50 \mathrm{m} .$ At point $\mathbb{B}$ , the blue bead collides elastically with a $0.600-\mathrm{kg}$ green bead at rest. Find the maximum height the green bead rises as it moves up the wire.

Robert Daine

Numerade Educator

Problem 64

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
Review. A metal cannonball of mass $m$ rests next to a tree at the very edge of a cliff 36.0 m above the surface of the ocean. In an effort to knock the cannonball off the cliff, some children tie one end of a rope around a stone of mass 80.0 kg and the other end to a tree limb just above the cannonball. They tighten the rope so that the stone just clears the ground and hangs next to the cannonball. The children manage to swing the stone back until it is held at rest 1.80 m above the ground. The children release the stone, which then swings down and makes a head-on, elastic collision with the cannonball, projecting it horizontally off the cliff. The cannonball lands in the ocean a horizontal distance R away from its initial position. (a) Find the horizontal component R of the cannonball’s displacement as it depends on m. (b) What is the maximum possible value for R, and (c) to what value of m does it correspond? (d) For the stone–cannonball–Earth system, is mechanical energy conserved throughout the process? Is this principle sufficient to solve the entire problem? Explain. (e) What if? Show that R does not depend on the value of the gravitational acceleration. Is this result remarkable? State how one might make sense of it.

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 65

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
Review. A bullet of mass m is fired into a block of mass Minitially at rest at the edge of a frictionless table of height $h(\text { Fig. } \mathrm{P} 9.65) .$ The bullet remains in the block, and after impact the block lands a distance $d$ from the bottom of the table. Determine the initial speed of the bullet.

Robert Daine

Numerade Educator

Problem 66

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
A small block of mass $m_{1}=0.500 \mathrm{kg}$ is released from rest at the top of a frictionless, curve-shaped wedge of mass $m_{2}=$ 3.00 $\mathrm{kg}$ , which sits on a frictionless, horizontal surface as shown in Figure $\mathrm{P} 9.66 \mathrm{a}$ . When the block leaves the wedge, its velocity is measured to be 4.00 $\mathrm{m} / \mathrm{s}$ to the right as shown in Figure $\mathrm{P} 9.66 \mathrm{b}$ . (a) What is the velocity of the wedge after the block reaches the horizontal surface? (b) What is the height $h$ of the wedge?

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 67

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
A $0.500-\mathrm{kg}$ sphere moving with a velocity given by
$(2.00 \hat{\mathrm{i}}-3.00 \hat{\mathrm{j}}+1.00 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s}$ strikes another sphere of mass 1.50 kg moving with an initial velocity of $(-1.00 \hat{\mathbf{i}}+2.00 \hat{\mathbf{j}}-3.00 \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s} .$ (a) The velocity of the $0.500-$ kg sphere after the collision is $(-1.00 \hat{\mathbf{i}}+3.00 \hat{\mathbf{j}}-8.00 \hat{\mathbf{k}})$
$\mathrm{m} / \mathrm{s}$ . Find the final velocity of the $1.50-\mathrm{kg}$ sphere and identify the kind of collision (elastic, inelastic, or perfectly inelastic). (b) Now assume the velocity of the 0.500 -kg sphere after the collision is $(-0.250 \hat{\mathbf{i}}+0.750 \hat{\mathbf{j}}-2.00 \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s}$ . Find the final velocity of the 1.50 -kg sphere and identify the kind of collision. (c) What If? Take the velocity of the $0.500-\mathrm{kg}$ sphere after the collision as $(-1.00 \mathrm{i}+3.00 \mathrm{j}+a \mathrm{k}) \mathrm{m} / \mathrm{s}$ . Find the value of $a$ and the velocity of the $1.50-\mathrm{kg}$ sphere after an elastic collision.

Robert Daine

Numerade Educator

Problem 68

A 75.0 -kg firefighter slides down a pole while a constant friction force of 300 $\mathrm{N}$ retards her motion. A horizontal 20.0 -kg platform is supported by a spring at the bottom of the pole to cushion the fall. The firefighter starts from rest 4.00 $\mathrm{m}$ above the platform, and the spring constant is
4000 $\mathrm{N} / \mathrm{m}$ . Find (a) the firefighter's speed just before she collides with the platform and (b) the maximum distance the spring is compressed. Assume the friction force acts during the entire motion.

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 69

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
George of the Jungle, with mass $m,$ swings on a light vine hanging from a stationary tree branch. A second vine of equal length hangs from the same point, and a gorilla of larger mass $M$ swings in the opposite direction on it. Both vines are horizontal when the primates start from rest at the same moment. George and the gorilla meet at the low- est point of their swings. Each is afraid that one vine will break, so they grab each other and hang on. They swing upward together, reaching a point where the vines make
an angle of $35.0^{\circ}$ with the vertical. Find the value of the ratio $m / M .$

Robert Daine

Numerade Educator

Problem 70

Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
Review. A student performs a ballistic pendulum experiment using an apparatus similar to that discussed in Example 9.6 and shown in Figure P9.56. She obtains the following average data: $h=8.68 \mathrm{cm},$ projectile mass $m_{1}=$ 68.8 $\mathrm{g}$ , and pendulum mass $m_{2}=263 \mathrm{g}$ . (a) Determine the initial speed $v_{1 A}$ of the projectile. (b) The second part of her experiment is to obtain $v_{1 A}$ by firing the same projectile horizontally (with the pendulum removed from the path) and measuring its final horizontal position $x$ and distance of fall $y$ (Fig. P9.70). What numerical value does she obtain for $v_{1 A}$ based on her measured values of $x=257 \mathrm{cm}$ and $y=85.3 \mathrm{cm}^{2}$ (c) What factors might account for the difference in this value compared with that obtained in part (a)?

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 71

Review. A light spring of force constant 3.85 N/m is compressed by 8.00 cm and held between a 0.250-kg block on the left and a 0.500-kg block on the right. Both blocks are at rest on a horizontal surface. The blocks are released simultaneously so that the spring tends to push them apart. Find the maximum velocity each block attains if the coefficient of kinetic friction between each block and the surface is
(a) 0, (b) 0.100, and (c) 0.462. Assume the coefficient of static friction is greater than the coefficient of kinetic friction in every case.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 72

Consider as a system the Sun with the Earth in a circular orbit around it. Find the magnitude of the change in the velocity of the Sun relative to the center of mass of the system over a six-month period. Ignore the influence of other celestial objects. You may obtain the necessary astronomical data from the endpapers of the book.

Sheh Lit Chang

University of Washington

Problem 73

A 5.00-g bullet moving with an initial speed of $v_{i}=400 \mathrm{m} / \mathrm{s}$ is fired into and passes through a 1.00 -kg block as shown in 1.00 -kg block as shown in Figure P9.73. The block, initially at rest on tal surface, is connected to a spring with force constant $900 \mathrm{N} / \mathrm{m} .$ The block moves $d=5.00 \mathrm{cm}$ to the right after impact before being brought to rest by the spring. Find (a) the speed at which the bullet emerges from the block and (b) the amount of initial kinetic energy
of the bullet that is converted into internal energy in the bullet–block system during the collision.

Robert Daine

Numerade Educator

Problem 74

Review. There are (one can say) three coequal theories of motion for a single particle: Newton’s second law, stating that the total force on the particle causes its acceleration; the work–kinetic energy theorem, stating that the total work on the particle causes its change in kinetic energy; and the impulse-momentum theorem, stating that the total impulse on the particle causes its change in
momentum. In this problem, you compare predictions of the three theories in one particular case. A $3.00-\mathrm{kg}$ object has velocity 7.00$\hat{\mathrm{j}} \mathrm{m} / \mathrm{s}$ . Then, a constant net force 12.0 $\mathrm{i} \mathrm{N}$ acts on the object for 5.00 s. (a) Calculate the object's final velocity, using the impulse-momentum theorem. (b) Calculate its acceleration from $\overrightarrow{\mathbf{a}}=\left(\overrightarrow{\mathbf{v}}_{f} \overrightarrow{\mathbf{v}}_{i}\right) / \Delta t$ . (c) Calculate its acceleration from $\overrightarrow{\mathbf{a}}=\Sigma \mathbf{F} / m$ . (d) Find the object's vector displacement from $\Delta \mathbf{r}=\overrightarrow{\mathbf{v}}_{i} t+\frac{1}{2} \overrightarrow{\mathbf{a}} t^{2} .$ (e) Find the work done on the object from $W=\overrightarrow{\mathbf{F}} \cdot \Delta \overrightarrow{\mathbf{r}} .$ (f) Find the final kinetic energy from $\frac{1}{2} m v_{f}^{2}=\frac{1}{2} m \overrightarrow{\mathbf{v}}_{f} \cdot \overrightarrow{\mathbf{v}}_{f} .$ (g) Find the final kinetic energy from $\frac{1}{2} m v_{i}^{2}+W .$ (h) State the result of comparing the answers to parts $(\mathrm{b})$ and $(\mathrm{c}),$ and the answers to parts $(\mathrm{f})$ and $(\mathrm{g}) .$

Sheh Lit Chang

University of Washington

Problem 75

Two particles with masses $m$ and 3$m$ are moving toward each other along the $x$ axis with the same initial speeds $v_{i}$ . Particle $m$ is traveling to the left, and particle 3$m$ is traveling to the right. They undergo an elastic glancing collision such that particle $m$ is moving in the negative $y$ direction
after the collision at a right angle from its initial direction. (a) Find the final speeds of the two particles in terms of $v_{i}$ . (b) What is the angle $\theta$ at which the particle 3$m$ is scattered?

Vishal Gupta

Numerade Educator

Problem 76

In the 1968 Olympic games, University of Oregon jumper Dick Fosbury introduced a new technique of high jumping called the “Fosbury flop.” It contributed to raising the world record by about 30 cm and is currently used by nearly every world-class jumper. In this technique, the jumper goes over the bar face up while arching her back as much as possible as shown in Figure P9.76a. This action places her center of mass outside her body, below her back. As her body goes over the bar, her center of mass passes below the bar. Because a given energy input implies a certain elevation for her center of mass, the action of arching her back means that her body is higher than if her back were straight. As a model, consider the jumper as a thin uniform rod of length L. When the rod is straight, its center of mass is at its center. Now bend the rod in a circular arc so that it subtends an angle of 90.08 at the center of the arc as shown in Figure P9.76b. In this configuration, how far outside the rod is the center of mass?

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 77

On a horizontal air track, a glider of mass m carries a G-shaped post. The post supports a small dense sphere, also of mass $m,$ hanging just above the top of the glider on a cord of length $L$ . The glider and sphere are initially at rest with the cord vertical. (Figure P9.47a shows a cart and a sphere similarly connected.) A constant horizontal force of magnitude $F$ is applied to the glider, moving it through displacement $x_{1} ;$ then the force is removed. During the time interval when the force is applied, the sphere moves through a displacement with horizontal component $x_{2} .$ (a) Find the horizontal component of the velocity of the center of mass of the glider-sphere system when the force is removed. (b) After the force is removed, the glider continues to move on the track and the sphere swings back and forth, both without friction. Find an expression for the largest angle the cord makes with the vertical.

Sheh Lit Chang

University of Washington

Problem 78

Sand from a stationary hopper falls onto a moving conveyor belt at the rate of 5.00 kg/s as shown in Figure P9.78. The conveyor belt is supported by frictionless rollers and moves at a constant speed of $v=0.750 \mathrm{m} / \mathrm{s}$ under the action of a constant horizontal external force $\overrightarrow{\mathbf{F}}_{\text { ext }}$ Supplied by the motor that drives the belt. Find (a) the sand's rate of change of momentum in the horizontal direction, $(\mathrm{b})$ the force of friction exerted by the belt on the sand, $(\mathrm{c})$ the external force $\overrightarrow{\mathbf{F}}_{\mathrm{ext}},(\mathrm{d})$ the work done by $\overrightarrow{\mathbf{F}}_{\mathrm{ext}}$ in $1 \mathrm{s},$ and (e) the kinetic energy acquired by the falling sand each second due to the change in its horizontal motion. (f) Why are the answers to parts (d) and (e) different?

Learnmore Shenje

Learnmore Shenje

Numerade Educator

Problem 79

Review. A chain of length L and total mass M is released from rest with its lower end just touching the top
of a table as shown in Figure P9.79a. Find the force exerted by the table on the chain after the chain has fallen through a distance x as shown in Figure P9.79b. (Assume each link comes to rest the instant it reaches the table.)

Sanjeev Kumar

Numerade Educator