Physics 2210 Paolo Gondolo

Fall 2005

FINAL EXAM -- REVIEW PROBLEMS

A solution set is available on the course web page in pdf format. A data sheet is provided.

1. A rock is thrown downward from the top of a building 175 m high with

a initial velocity of 37.0 m/s. At the same time another rock of equal

mass is projected upward from the bottom of the building with an initial

speed of 30.0 m/s.

(a) Calculate the height above the ground at which the two rocks

meet.

(b) If the rocks collide in a completely inelastic collision, calculate

the time when they reach the ground. Let t = 0 be the time when

the rocks are initially launched.

2. A small solid ball of mass m and radius r is started at rest

and rolls without friction down the loop-the-loop shown.

Assume r << R. (h = 4R).

(a) Calculate the normal force of the track on the ball at

point A, exactly opposite the center of the loop.

(Your answer should be expressed as the weight of

the ball times a number.)

(b) Calculate the normal force on the ball at point B,

exactly at the top of the loop. (Your answer should

be expressed as the weight of the ball times a

number.)

3. A small block of mass m is placed on a conical surface that is rotated

about its symmetry axis. The coefficients of friction are 0.65 and 0.55.

The block is initially a distance d up the cone as shown. (m = 0.750 kg,

d = 0.374 m)

(a) Calculate the maximum possible angular speed for rotation of the

cone such that the block does not slide. Free body and force

diagrams are an essential part of this problem.

(b) At the instant before the block starts to slide, calculate the

frictional force and the normal force.

4. (a) Two cylinders of mass m = 2.45 kg and radius R = 3.50 cm

each, are connected by a massless rod of length 5.00 cm. This is

shown in cross section. The system is rotated about an axis

perpendicular to the paper and through the center of the rod. Find

the moment of inertia of the system.

(b) A grinding wheel, in the shape of a disc of thickness 1 inch and

radius 12 inches, is slowed by friction with the tool being ground. The weight of the wheel is 55

pounds. If it slows from 25 rad/s to a full stop in 37 seconds, find the frictional force exerted by the

tool.

(c) Find the weight on Earth of an object whose mass on the moon is 2.70 kg.

(d) If the coefficient of friction between skis and snow is 0.075, find the angle of slope at which a skier

will move with constant velocity.

(e) A 4.75 kg mass is rotated on a string. The string rotates in a cone, where the angle of the string

from the vertical is 27.0°. Find the tension in the string.

5. A hollow cylinder is rotating about a vertical axis with angular velocity T, as

shown. A small block of mass m is on the inside of the cylinder. A string exerts

a constant force P equal to 3 times the weight of the block in a direction of 30°

from the vertical, as shown. The coefficients of friction between the block and

kthe wall are : = 0.60. The radius R is 1.25 m, and the block is very small.

Find the minimum angular velocity of the cylinder such that the block does not

slide up.

6. At t = 0 a rock is dropped from the top of a building 150 m high. Exactly

o2.00 s later another rock is thrown down with an initial velocity of v m/s.

(a) Calculate the initial velocity for the second rock, such that both rocks

arrive at the bottom at the same instant.

(b) Calculate the velocity of the thrown rock when it reaches the bottom.

7. A mass m is on the slope of a cone that rotates about its symmetry

axis. There is a string attached to the mass and is fixed at the

center. The string is parallel to the slope of the cone. The mass m

does not move with respect to the cone. The tangential velocity of

the mass (into the paper) is 2.55 m/s. Clear free body and force

diagrams are a necessary part of this problem.

(a) If the string stretches slightly so that the maximum static

friction occurs between the cone and the mass, calculate the

tension in the string. The diagram shows the position and

dimensions with the string stretched.

(b) Calculate the frictional force acting on the mass.

s km = 0.325 kg; 2 = 25.0°; R = 0.65 m; : = 0.650; : = .550

S K8. A car is traveling down a 15.0° slope as shown. The coefficients of friction are : = 0.65, : = 0.57.

(a) If the car is going at 55.0 mi/hr, how far would it skid before stopping with wheels locked?

(b) For the same initial conditions, how far would the car skid if it is initially going uphill?

(c) For part (a), calculate the time to reduce the speed of 1/2 of its initial value.

9. A cylinder of radius R (not small) and mass M rolls

without sliding on a surface with the shape shown. It starts

from rest.

(a) Calculate the largest possible value of h, such that

the cylinder does not leave the surface when it

passes over the hump. h is measured to the center of

mass of the cylinder. Express this in terms of R and

A (the radius of the

top of the hump). The top of the hump is 2A above the ground.

(b) For the starting value of h calculated in (a), the apparent

weight of the cylinder at C is found

to be 4 Mg. Find the radius of curvature at C.

10. A wood block of mass m = 2.05 kg is dropped from rest from

the top of a 150 m cliff on Earth at t = 0. After it has fallen

for 2.00 s, a bullet is fired upwards from the ground. The

initial speed of the bullet is 115 m/s (at the ground). The mass of the bullet is 30.0 g.

(a) Find the height above the ground where the bullet strikes the block.

(b) If the bullet sticks in the block, find the time (measured from t = 0) when the block hits the ground.

11. (a) Four cylinders are touching each in the arrangement shown.

Calculate the moment of inertia for rotation about the axis A, at the

exact center, and perpendicular to the paper. Take the mass of each

cylinder as M, and the radius of each as R.

(b) A wheel is accelerated from rest at an angular acceleration of 3.75 r/s . Calculate the total angular2

displacement after 8.90 s.

(c) For the wheel in (b), calculate the magnitude of the angular velocity when the total angular

displacement is 18.0 rad.

(d) A car is traveling at 60 mi/hr. If the wheels have a radius of 15 inches, what is the magnitude of

their angular velocity in rad/s?

(e) A bowling ball has a weight of 16.0 pounds. If it is rolling without sliding and the radius is 4.00

inches, calculate the rotational kinetic energy if its translational speed is 30.0 ft/s.

(f) Calculate magnitude of the angular momentum of a baseball of mass 0.145 kg if it rotates at 1600

RPM (revolutions per minute). Assume a uniform mass distribution. Take the radius as 4.00 cm.

12. Car A passes the point x = 0 at a steady speed of 75.0 mi/hr.

After A has traveled 250 feet a police car, P, starts from rest, and

uniformly accelerates. Car P catches car A at point x = 2000 ft.

(a) Calculate the acceleration of P.

(b) Calculate the velocity of P when P catches A.

13. Mass 1 rests on a rotating turntable (like our green lazy susan) a

distance d from the axis of rotation. A string from 1 passes over a

massless, frictionless pulley, and is attached to mass 2. By means of

2a mechanism not shown, and not relevant to the problem, m is

constrained to move only in the vertical direction. (It rotates around

with the lazy susan also.)

1 2 sd = 0.300 m; m = 1.27 kg; m = 0.300 kg; : = 0.650

(a) Calculate the maximum value of the tangential velocity due to

1 1rotation of m , into the paper, such that m does not slide.

2(b) Calculate the maximum value of m for which this is a

sensible problem. (Find an analytic expression for the

2velocity in a, and see what values of m do not work.)

14. A block of mass m is given an initial velocity by the spring. The spring has a

spring constant k, and is squeezed 0.200 m when the block is at A. Between A

s kand B the slope has friction coefficients : = 0.75 and : = 0.65. After point B

the system is frictionless. The block goes around the circular loop of radius R

and is in contact with the loop all the way. The distance between A and B is

2.40 m. Assume the size of the block is small compared to R. The vertical

position of B is the same as the center of the loop.

m = 0.175 kg; k = 120.0 N/m; R = 0.500 m; 2 = 27.0°

(a) Calculate the speed of the block at the exact top of the loop.

(b) Find the normal force of the loop on the block at the exact top.

(c) Determine the normal force on the block at C, directly opposite the center of the loop.

15. A policeman (P) observes a car (A) going by. Five (5.00 ) seconds

after car A goes by, the policeman starts off at a uniform acceleration

of 3.00 m/s . The policeman catches car A 16.0 s after A passes2

him.

(a) How fast was the policeman going when he catches up to car

A?

(b) How far is the policeman from his starting point when he

catches car A?

(c) How fast was car A going? (Assume he maintains a steady

speed.)

16. The system shown is released from rest. The pulley is frictionless and

massless

(a) Determine which direction the system moves. The arrow shows

the positive direction.

(b) Calculate the acceleration of the system.

(c) What is the speed after block 1 has fallen 1.50 m?

(d) How much time does it take for the system to move 2.25 m?

17. A bomber is flying horizontally at a speed of 275 m/s at an

altitude of H = 3000 m. What distance, R, from a target must a

bomb be released in order to hit the target? The distance R ismeasured by a laser scope straight from the plane to the target.

The target is mall. Neglect air resistance.

18. A cone with " = 40.0° is rotating about the y axis with an

angular velocity of T = 3.00 rad/s. Calculate the range of heights

min max(y , y ) where you can place a block of mass M = 3.50 kg so

that it would stay there (that is, it would not slide up or down).

s: = 0.41

k: = 0.35

19. (a) A car with wheels of diameter 28.0 inches is traveling at 60.0 mi/hr. Calculate the angular velocity

of the wheels in rad/s.

(b) Two cars are each traveling with a speed of 30.0 mi/hr. They

collide head on. The cars each have a mass of 1500 kg. The

collision is completely inelastic. What percentage of the initial

kinetic energy is lost in this collision?

(c) Two solid spheres of radius 5.25 cm and mass 0.560 kg, are touching.

Calculate the moment of inertia for rotation about the axis that goes through

the point that they touch and is tangent to the spheres at that point.

(d) A solid sphere of mass 1.25 kg and radius 3.00 cm is rotating at 72.0 rad/s

about an axis through its center of mass. It slows to a complete stop in

exactly 175 revolutions. Assuming the acceleration is constant, what is the torque acting on the

sphere?

(e) For the object shown the rod is massless and the

masses are small. Take a = 1.25 m. Calculate

the moment of inertia for rotation about the axis

shown.

(f) A cannon is fired horizontally with a velocity of

750 m/s from a cliff on the moon. Find the distance x

from the base of the cliff that the cannon ball lands.

Ignore any curvature of the moon’s surface.

20. A block is at rest on an inclined plane. The external force F is

applied horizontally.

(a) Calculate the maximum value of F such that the block

does not move.

(b) Find the acceleration of the block if the external force is

F = 65.0 N.

s km = 3.20 kg; : = 0.75; : = 0.60

21. The car on a frictionless roller coaster starts at rest at

position A. The hump at B has a radius of curvature R.

B is 2R above the ground level.

(a) Calculate the value of R (in terms of h) such that

the normal force of the track on the car at B is

exactly half of the weight of the car.

(b) For the same starting location and value of R,

find the maximum initial velocity needed at A

such that the normal force at B is zero.

22. The block shown is launched down the incline with a speed of

5.25 m/s. Its mass is 0.850 kg. It travels 1.50 m and strikes

a spring that is at its equilibrium length. The spring constant

is k = 420 N/m.

(a) Find the maximum compression of the spring (in cm).

(b) If the spring is compressed 20.0 cm with the block in

contact and then released,, calculate how far up the

incline the block will go. Measure from point A, the

equilibrium position of the end of the spring.

s km = 0.850; : = 0.70; : = 0.55

23. Block 1 is launched up the frictionless plane at an initial velocity

oof v = 1.25 m/s. Block 2 is released from rest at the same time

as block 1 is launched.

(a) Find the location of the collision between blocks 1 and 2.

Measure this up the plane from the initial position of block

1. Assume the blocks are small.

(b) If the collision between blocks 1 and 2 is completely

inelastic, find the velocity after the collision. Take up the

plane as positive. If you cannot do (a), do (b) symbolically.

1 2 om = 4.30 kg; m = 2.75 kg; v = 1.25 m/s; d = 6.00 m

24. A block of mass M = 5.00 kg is supported by two string

which are wrapped around a cylinder of m = 40.0 kg and

R = 20.0 cm. The cylinder is on a horizontal axis

without friction. The system is released from rest.

(a) Find the velocity v of the block after it falls a

distance h = 5.00 m.

(b) What is the acceleration of the block?

(c) How long does it take the block to fall the distance

h = 5.00 m?

25. A small block of mass m is launched on the frictionless loop as

shown. The spring launcher has a spring constant of k.

(a) Find the velocity of the block at the top of the loop (point

A) as a function of initial spring compression x.

(b) Determine the minimum value of x such that the block goes

over the loop in contact with the loop.

(c) Using the value of x determined in (b), calculate the

x yhorizontal and vertical components of all forces (F , F )

exerted on the block at point C.

26. (a) A disc of mass M and radius R rotates on an axle through its center. Around

the edge are three smaller discs of radius 1/3 R and mass 1/9 M. Their

centers are at the exact edge of the main disc. Calculate the moment of

inertia of this system for rotation about the center of the main disc (shown

by A).

(b) The planet Jupiter has a mass of approximately 1.80 × 10 kg. Its moon, Io,27

has an orbit about the planet with a period of 42.0 hours. Find the distance from Io to the center of

Jupiter from this data assuming the orbit of Io is circular.

(c) Calculate the gravitational force between two spherical objects whose centers are 5.00 cm apart. One

has a mass of 1.00 kg and the other a mass of 0.100 kg.

(d) A traveling wave is described by the function y = [3.25 × 10 m] cos (12.0 x - (1.50 × 10 )t). -3 3

Determine the speed of this wave.

(e) Calculate the period, on the moon, of a simple pendulum 3.00 m long.

(f) If the mass of the earth were doubled and the radius of the earth were doubled, what would be the new

value of "g"?

27. The two blocks shown are on an inclined plane. The force F is parallel to the

plane. All surfaces have the same coefficients of friction.

(a) Draw careful, clear, free body and force diagrams for object 1 and object

2. Label them clearly.

(b) Calculate the maximum value of F such that object 2 does NOT slide with

respect to object 1.

1 2m = 3.25 kg m = 2.65 kg

s k: = 0.60 : = 0.40

2 = 27.0°

28. A small sphere (r << R), of mass m, rolls without sliding on the

loop-the-loop shown. It starts with zero velocity at point A, a

distance 9R above the bottom of the loop. The loop is circular.

(a) What is the speed of the center-of-mass of the sphere at

point B? B is at the same level as the center of the loop.

(b) Find the normal force on the sphere at B.

(c) Find the normal force on the sphere at point C, the exact

top.

29. Block 1 is at rest on the spring after all motion has ceased. Block 2 is dropped

from a height h above block 1 and they collide in a completely inelastic collision.

The spring is long enough that its length is not an issue.

(a) Calculate the speed of blocks 1 and 2 the instant after the collision.

(b) Using y = 0 as shown in the diagram, find the value of y (for the bottom of

block 1) when the spring is at its maximum compression.

(c) What is the angular frequency of the oscillation of this system after the

collision?

(d) Find the value of y (the bottom of block 1) when the system comes to rest

after all oscillations have died out.

1 2m = 4.75 kg m = 2.50 kg

h = 1.75 m k = 1100 N/m

30. Block 2 slides on a frictionless table. The pulley is a cylinder whose radius is

3.00 cm and its mass is 2.20 kg. It turns on a frictionless axle. The rope does

NOT slip on the cylinder.

(a) Using energy methods, calculate the speed of block 2 after it has moved

0.65 m from rest.

(b) Find the acceleration of block 2.

1 2m = 3.40 kg m = 7.95 kg

31. Initially block 2 is at rest and the spring is at its equilibrium length. The two

masses are on a frictionless table. Mass 1 is launched with an initial velocity

ov , and collides in a completely inelastic collision with Mass 2.

(a) Find the frequency f, and the angular frequency T, for the resulting

oscillations.

(b) Write a complete expression describing the oscillations in the form x = A cos (Tt ! N), and evaluate A,

T and N numerically, including the sign in front of N.

1 2m = 4.35 kg m = 6.75 kg

ok = 750 N/m V = 4.00 m/s

32. The wave velocity on a violin string is given by

, where T is the tension in Newtons, and :

the linear mass density in kg/m. A violin string is tuned so

that the 2 overtone (two nodes between the supports, asnd

shown) has a frequency of 1400 Hz.

(a) Calculate the frequency of the fundamental.

(b) Find the frequency of the 4 overtone (4 nodes between supports).th

(c) The tension is increase by 5.00%. Nothing else is changed. What is the new frequency of the 2nd

overtone (the one pictured above)?

33. (a) On a small planet a stone is dropped. It falls 6.0 ft and hits the ground with a velocity of 0.752 m/s.

Fin g on this planet.

(b) The stopping distance for a car traveling at 60.0 mi/hr is 70.0 ft. What is the stopping distance of

the same car traveling at 20.0 mi/hr?

(c) A car on a circular track of 200 m radius is traveling at 145 mi/hr at point A. It

slows down with uniform acceleration so that at point B its speed is 90 mi/hr. A

and B are 90° apart. Calculate the tangential acceleration of the car along the

track between points A and B. (Hint: Use 1-D kinematics along the curved

path.)

(d) The moon orbits the earth in 27 1/3 days at a distance of 240,000 miles. Assume the mass of the moon

is small compared to the mass of the earth. Calculate the inward acceleration of the moon in m/s 2

(e) A spring follows the force law F = -kx . If k = 5.60 N/m , find the energy stored in the spring when it4 4

is expanded by 0.250 m from its equilibrium position.

Fig. 1a Fig. 1b

34. (a) Find the center of mass of the system shown.

(b) On a frictionless air track two carts, each with a mass of 2.30 kg, are moving in opposite directions

toward each other. Each cart has a speed of 0.250 m/s. If the two carts collide and stick together, how

much mechanical energy is lost to heat?

(c) Find the kinetic energy of a uniform solid disk of mass 2.75 kg and radius 0.200 m rotating about its

center at 3050 revolutions per hour.

(d) Two spheres touch each other. Each has a mass of 5.00 kg and a radius of 4.25

cm. Calculate the moment of inertia for rotation about an axis through the

point of contact and tangent to the two spheres, as shown.

(e) A 0.205 kg mass on a horizontal frictionless surface oscillates according to the

following function x = 7.15 cos (3.05 t + 20.0°), where x is measured in meters and the frequency is

measured in rad/s. Calculate the energy (kinetic plus potential) in the oscillations.

35. A particle of mass : is in a circular orbit of radius r around a star of mass M.

(a) What is the total (kinetic plus potential) energy of the particle?

(b) What is the total angular momentum of the particle?

(c) Suppose that a drag force acts on the particle and the particle very gradually spirals inward on an

orbit that is always approximately circular. The direction of the drag force is opposite to the

direction of the velocity of the particle. Does the particle increase in speed, or decrease in speed and

why? You must justify your answer.

36. The object shown is bounded by the horizontal x-axis, the line x = x

o, and the curve x = ay . It has a thickness t, and a density D. 3

Calculate the x-coordinate of the center of mass of this object.

37. A solid flywheel, which has a mass of 10 kg and a radius of 5 m, is5

rotating at 200 rpm (Fig. 1a). Consider the flywheel to be a solid disk of constant mass density.

(a) The flywheel is brought into contact with a rough surface that provides a frictional force of 20 kN

(Fig. 1b). Find the time it takes flywheel to stop.

(b) If the flywheel rolls down a 30° slope starting from rest, how far down the slope must the flywheel

roll (without sliding) to reach the same angular velocity of 200 rpm? [It might be easier to do this

problem using conservation of energy.]

38. (a) If I weigh 80.0 pounds on the moon, what are my weight and mass on the earth (be sure to supply the

correct units).

(b)

Calculate

(c) Find the moment of inertia about the point A for the system in the figure. The body on

the left is a solid cylinder of mass M and the body on the right is a hollow cylinder with

thin walls of mass M. The axis of rotation is at the exact center of the solid cylinder and

parallel to its long axis.

(d) A ball of mass m = 1.00 kg and moving with a velocity of = 2.00 m/s strikes a stationary ball of

mass 2 m. The collision is perfectly elastic. See diagram at right. If the final speed of the ball whose

1mass is m is v = 1.00 m/s, what is the speed of the ball whose mass is 2 m?

(e) A wheel that rolls without slipping on level ground is accelerated from rest with an angular

acceleration of 5.00 rad/s . The radius of the wheel is 1.00 m. How far has the center of the wheel2

traveled after 9.00 s, in meters?

.39 (a) A solid sphere of mass 2.00 kg with r = is spinning on a fixed axis through

its center at 2.00 revolutions per second. What is its angular momentum?

(b) An elevator that weighs 2.00 × 10 pounds is pulled upward with a constant acceleration of 2.00 m/s . 3 2

What is the tension in the elevator cable?

(c) A solid cylinder has a mass of 1.50 kg and a radius of 3.00 cm. The cylinder is rotating about

its long axis. At point (A) on the surface of the cylinder, is moving at a speed of 10.0 m/s.

What is the rotational kinetic energy of the cylinder?

(d) Consider the gyroscope shown in the figure. Draw the angular momentum

vector, the net force vector and the torque vector.

(e) A wave of angular frequency 25.0 rad/s has a wave vector k = 10.0 cm . What is the velocity of this-1

wave?

40. The block shown has a mass of 1.58 kg. The block is pulled up the incline

sby an external force F, as shown. The coefficients of friction are : =

k0.700 and : = 0.550. The force F is 7.50 N. The block stays in contact

with the plane at all times. If the block is moved 2.25 m up the incline,

calculate (including signs):

(a) the work done by gravity on the block;

(b) the work done on the block by the normal force;

(c) the work done by F on the block;

(d) the work done by friction on the block.

Fig. 6a Fig. 6b

41. A 5.80 kg block rests on a 35.0° slope and is attached by a string of negligible mass to a

solid drum (cylinder) of mass 2.00 kg and radius 10.0 cm, as shown in the figure. When

released, the block accelerates down the slope at 2.30 m/s . What is the coefficient of2

friction between the block and slope?

42. A 1.00 kg block slides down the plane shown in the figure. The initial

velocity of the block is 5.00 m/s. The maximum value of the coefficient of

sstatic friction is : = 0.800

k(a) If the coefficient of dynamic friction is : = 0.600, how far down the

plane will the block go before stopping?

s(b) After the block stops, what is the value of the force of friction, F ?

(c) If the angle of the plane is increased to 40°, will the block stop? You

must give a valid reason for your answer?

143. A mass m is attached to a wire of linear density 5.60 g/m, and the other end of the wire run over a pulley and

tied to a wall as shown in Fig. 6a. The speed of the transverse waves on the horizontal section of the wire is

2observed to be 20.0 m/s. If a second mass m is added to the first, the wave speed increases to 45.0 m/s. See

Fig. 6b. Find the second mass. Assume the string does not stretch.

44. A. A train accelerates at 2.0 m/s for 1.0 s while moving a distance of 3.0 m. What is its final speed in2

m/s? Caution: its initial speed is not zero.

B. A uniform plank of mass M = 4.0 kg and length L = 1.2 m is pivoted at one end.

The plank’s other end is supported by a spring of force constant k = 85 N/m (see

figure). In equilibrium the plank is horizontal and the spring makes an angle

q = 30° with the plank. What is the elongation of the spring?

45. A. An object of mass M is rotating about a fixed axis with angular momentum L about

the axis. Its moment of inertia about the axis is I. What is its kinetic energy? To

get full credit, you must show your reasoning.

(a) IL /2 .2

(b) L /2I .2

(c) ML /2 .2

(d). IL /2M .2

B. Two children, Andrew and Barbara, are skating on a frozen lake. They slide

toward each other, grab their hands, and spin together gliding, as sketched in

the figure. The children weigh m = 35 kg each, and approach each other at a

distance d = 1.2 m with the same initial speed v = 2.0 m/s. Treating the

skaters as point particles, and neglecting friction and air resistance, find how

many turns they spin in half a minute.

46. A 3.5 kg cat is sliding down a wet slide accelerating at 1.1 m/s . The slide makes an angle of 45° with the2

horizontal.

(a) Draw a force diagram for the cat.

(b) Find the coefficient of kinetic friction between the cat and the slide.

47. A wooden block of mass M = 100 g is attached to a spring of spring

constant k = 9.0 N/m as shown in the figure. The block is initially at rest

on a frictionless table. A bullet of mass m = 20 g is fired horizontally at

a speed v = 4.2 m/s. The bullet gets stuck inside the block. Find

(a) the velocity of the block with bullet just after the impact,

(b) the energy of the block with bullet just after the impact, and

(c) the maximum compression of the spring.

48. Your physics professor has decided to take up snowboarding at

Snowbird. After some initial success, he gets stuck in a half-pipe

and keeps oscillating back and forth near its bottom, rigid with

panic. Consider the professor as the weight of a physical pendulum

oscillating about an imaginary pivot P at the center of the half-pipe

(see figure). Let the radius of the half-pipe be r= 2.0 m, and model

the professor as a uniform cylinder of diameter D = 30 cm, height

H = 160 cm, and mass M = 75 kg. Neglect the mass of the

snowboard, and neglect friction.

(a) Find the moment of inertia of the cylinder that models your

professor about the imaginary pivot P.

(b) Find the period of your professor’s harmonic oscillations.

49. A. Suppose you swing on a swing standing on it instead of sitting on it. Is your frequency of oscillation

larger when you sit or when you stand? Neglect the effect of air resistance. Explain your reasoning.

B. A satellite is in an elliptical orbit around Earth. What is the total amount of work done on the satellite

by the gravitational force of Earth during one complete orbit? Explain your reasoning.

C. A ball is rolling on the floor with angular velocity T. It then hits a vertical wall and bounces back.

cmThe ball has radius R and moment of inertia I about its center of mass. Assume there is no friction

cmbetween ball and wall and that the bounce is elastic. Express your answers in terms of T, R and I .

1. In which direction and at what angular speed is the ball spinning right after it bounces?

Explain your reasoning.

2. Right after the bounce, is the ball rolling or sliding? Explain your reasoning.

50. A 1.00 kg steel ball and a 2.00 m cord of negligible mass make up a simple

pendulum that can pivot without friction about the point O, as in the figure.

This pendulum is released from rest in a horizontal position. When the ball is

at its lowest point, it strikes a 6.00 kg block sitting at rest at the middle of a

2.00 m long shelf. The block starts sliding along the shelf, and the pendulum

swings back up until the cord makes an angle of 60° with the vertical.

(a) What is the speed of the ball when it hits the block?

(b) What is the velocity of the block just after impact? Give both magnitude

and direction.

(c) Determine if the collision is elastic or inelastic by comparing the total

kinetic energy before and after the collision. If inelastic, compute the

amount of kinetic energy lost.

51. The plunger of a pinball machine is used to launch a ball along

p the surface of a table. The plunger has mass m = 45.0 g and

is attached to a spring of force constant k = 30.0 kN/m (see

0figure). The spring is compressed a distance x = 1.50 cm

from its equilibrium position x = 0 and released. The ball has

bmass m = 30.0 g and is initially next to the plunger. Assume

that the surface is horizontal and frictionless so that the ball

slides without rolling.

(a) What is the position x of the plunger when the ball

separates from it?

f(b) At what distance x does the plunger come to rest

momentarily?

(c) What is the speed of the ball when it separates from the plunger?

(d) The ball eventually falls off the table from a height of 80.0 cm. At what distance from the edge of

the table does the ball hit the floor?

52. A constant-density cylinder of mass 0.50 kg and radius 4.0 cm can

rotate freely about an axis through its center. It has a thin metal wire

wound around an attached axle of radius 0.50 cm that also runs

through its center (see figure). The wire is attached to a mass of 1.5

kg, which slides down an inclined plane with an acceleration of 0.10

m/s . The straight portion of the wire vibrates at its fundamental2

frequency. The mass density of the wire is 0.20 g/m.

(a) What is the tension in the wire?

(b) What is the frequency at which the straight portion of the wire

vibrates when its length is L = 0.80 m? Use your result from

part (a) to determine the speed of waves in the wire.

(c) What is the coefficient of kinetic friction between the block and the plane?

Data: Use these constants (where it states for example, 1 ft, the 1 is exact for significant figure purposes).

1 ft = 12 in (exact)

1 m = 3.28 ft

1 mile = 5280 ft (exact)

1 hour = 3600 sec = 60 min (exact)

1 day = 24 hr (exact)

earthg = 9.80 m/s2

= 32.2 ft/s2

m oong = 1.67 m/s2

= 5.48 ft/s2

1 year = 365.25 days

1 kg = 0.0685 slug

1 N = 0.225 pound

1 horsepower = 550 ftApounds/s

earth(exact) M = 5.98 × 10 kg24

earthR = 6.38 × 10 km3

sunM = 1.99 × 10 kg30

sunR = 6.96 × 10 m8

moonM = 7.35 × 10 kg22

moonR = 1.74 × 10 km3

G = 6.67 × 10 N@m /kg-11 2 2

electronm = 9.11 × 10 kg-31