# Working with the motion of rigid bodies

1. During a violent thunderstorm, hail of diameter 1.0 cm falls directly downward at speed of 25m/s. There are estimated to be 120 hailstones per cubic meter of air. (a) what is the mass of each hailstone ( density = 0.92 g/cm^3) ? (b) Assuming that the hail does not bounce fine the magnitude of the average force on a flat roof measuring 10m x 20m due to the impact of the hail. ( Hint : during impact the force on a hailstone from the roof is approximately equal to the net force on the hailstone because the gravitational force on it is small.

2. Figure below shows an approximate plot of force magnitude versus time during the collision of a 58 g Superball with a wall. The initial velocity of the ball is 34 m/s perpendicular to the wall; it rebounds directly back with approximately the same speed, also perpendicular to the wall. What is F max , the maximum magnitude of the force on the ball from the wall during the collision ?

3. A 2500 kg unmanned space probe is moving in a straight line at a constant speed of 300 m/s. Control rockets on the space probe execute a burn in which a thrust of 3000 N acts for 65.0 s. (a) what is the change in the magnitude of the probes linear momentum if the thrust is backward , forward or directly sideways ? (b) what is the change in kinetic energy under the same three conditions ? assume that the mass of the ejected burn products is negligible compared to the mass of the space probe.

4. Two titanium spheres approach each other head on with the same speed and collide elastically. After the collision one of the spheres whose mass is 300 g remains at rest (a) What is the mass of the other sphere ? (b) What is the speed of the two-sphere center of mass if the initial speed of each sphere is 2.0 m/s ?

5. A billiard ball moving at a speed of 2.2 m/s strikes an identical stationary ball a glancing blow. After the collision one ball is found to be moving at a speed of 1.1 m/s in a direction making a 60 degree angle with the original line of motion. (a) Fine the velocity of the other ball. ( b) Can the collision be inelastic , given this data ?

6. An object of mass m undergoes a one-dimensional impulse vector J, with the speed changing from v to u without any change in the direction of travel. Show that the work is 1/2 J ( u + v )

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1. During a violent thunderstorm, hail of diameter 1.0 cm falls directly downward at speed of 25m/s. There are estimated to be 120 hailstones per cubic meter of air. (a) What is the mass of each hailstone (density = 0.92 g/cm^3)? (b) Assuming that the hail does not bounce, find the magnitude of the average force on a flat roof measuring 10m x 20m due to the impact of the hail. (Hint: during impact the force on a hailstone from the roof is approximately equal to the net force on the hailstone because the gravitational force on it is small.

Mass/ Volume = Density

Hence mass = density * volume = 0.92 * volume

Assuming a spherical shape for each stone, v = (4/3) pi * r^3 = 4.19 cm^3

Mass = 0.92 * 4.19 = 3.85 g

Mass of 120 stones = 3.85 * 120 = 462 g = 0.462 kg (mass per cubic meter of air)

Area of the roof A = 200 m^2; on every meter square of area 0.462 kg of stone will fall. Hence the total mass of the stone falling on the roof of area 200 m^2 is m = 200 * 0.462 = 92.4 kg

We have average force = impulse/time = change in momentum/time = m (v1-v2)/t

Here the stone doesn't bounce, hence v2 = 0

Thus the average force F = 92.4 * 25/t = 2310 Newton/s

2. Figure below shows an approximate plot of force magnitude versus time during the collision of a 58 g Superball with a wall. The initial velocity of the ball is 34 m/s perpendicular to the wall; it rebounds directly back with ...

#### Solution Summary

A good set of practice problems involving different practical applications - All questions in the attachment are answered.

A set of problems on circular and rotational motion, centre of mass, moment of inertia, simple harmonic motion.

1. A bicycle travels 141 m along a circular track of radius 15 m. What is the angular displacement in radians of the bicycle from its starting position?

a. 1.0 rad

b. 1.5 rad

c. 3.0 rad

d. 4.7 rad

e. 9.4 rad

2. Which equation is valid only when the angular measure is expressed in radians?

See the attachment

3. A grindstone, originally rotating at 126 rad/s undergoes a constant angular acceleration so that it makes 20.0 rev in the first 8.00 s. What is its angular acceleration?

a. 0.313 rad/s2

b. 0.625 rad/ s2

c. 2.50 rad/ s2

d. 1.97 rad/ s2

e. 13.79 rad/ s2

4. A 0.254-m diameter circular saw blade rotates at a constant angular speed of 117 rad/s. What is the tangential speed of the tip of a saw tooth at the edge of the blade?

a. 29.7 m/s

b. 14.9 m/s

c. 9.46 m/s

d. 7.45 m/s

e. 2.17 m/s

5. A ball attached to a string starts at rest and undergoes a constant angular acceleration as it travels in a horizontal circle of radius 0.30 m. After 0.65 sec, the angular speed of the ball is 9.7 rad/s, what is the tangential acceleration of the ball?

a. 4.5 m/s2

b. 0.32 m/s2

c. 15 m/s2

d. 7.6 m/s2

e. 28 m/s2

6. Two motorcycles are riding around a circular track at the same angular velocity. One motorcycle is at a radius of 15 m; and the second is at a radius of 18 m. What is the ratio of their linear speeds, v2/v1?

a. 1.0

b. 0.83

c. 1.4

d. 0.71

e. 1.2

7. Which statement concerning a wheel undergoing rolling motion is true?

a. The angular acceleration of the wheel must be zero.

b. The tangential velocity is the same for all points on the wheel.

c. The linear velocity for all points on the rim of the wheel is nonzero.

d. The tangential velocity is the same for all points on the rim of the wheel.

e. There is no slipping at the point where the wheel touches the surface on which it is rolling.

8. A car is moving along a horizontal road at a constant velocity that is directed 45 degree south of east. What is the direction of the angular velocity of the wheels of the cart?

a. 45 degree south of west

b. 45 degree north of west

c. 45 degree south of east

d. 45 degree north of east

e. due east

9. Complete the following statement: When a net torque is applied to a rigid object, it always produces a

a. constant acceleration.

b. rotational equilibrium.

c. constant angular velocity.

d. constant angular momentum.

e. change in angular velocity.

10. A meter stick is pivoted at the 0.50-m line. A 3.0-kg object is hung from the 0.10-m line. Where should a 5.0-kg object be hung to achieve equilibrium?

a. 0.06-m line

b. 0.24-m line

c. 0.56-m line

d. 0.74-m line

e. A 5.0-kg object cannot be used to sustain equilibrium in this system.

11. Which of the following statements most accurately describes the center of gravity of an object?

a. It is the point where gravity acts on the object.

b. It is the point where all the mass is concentrated.

c. It must be experimentally determined for all objects.

d. It is the point on the object where all the weight is concentrated.

e. It is the point from which the torque produced by the weight of the object can be calculated.

12. Three objects are positioned along the x axis as follows: 4.4 kg at x = +1.1 m, 3.7 kg at x = -0.80 m, and 2.9 kg at x = -1.6 m. The acceleration due to gravity is the same everywhere. What is the distance from the location of the center of gravity to the location of the center of mass for this system?

a. zero meters

b. - 0.52 m

c. - 0.26 m

d. + 0.26 m

e. + 0.52 m

13. A 50 N?m torque acts on a wheel of moment of inertia 150 kg?m2. If the wheel starts from rest, how long will it take the wheel to make one revolution?

a. 0.33 s

b. 0.66 s

c. 2.4 s

d. 6.1 s

e. 10 s

14. Which one of the following statements concerning the moment of inertia I is false?

a. I may be expressed in units of kg?m2.

b. I depends on the angular acceleration of the object as it rotates.

c. I depends on the location of the rotation axis relative to the particles that make up the object.

d. I depends on the orientation of the rotation axis relative to the particles that make up the object.

e. e. Of the particles that make up an object, the particle with the smallest mass may contribute the greatest amount to I.

15. A string is wrapped around a pulley of radius 0.10 m and moment of inertia 0.15 kg?m2. The string is pulled with a force of 12 N. What is the magnitude of the resulting angular acceleration of the pulley?

a. 18 rad/s2

b. 0.13 rad/s2

c. 80 rad/s2

d. 0.055 rad/s2

e. 8.0 rad/s2

16. A 1.0-kg wheel in the form of a solid disk rolls along a horizontal surface with a speed of 6.0 m/s. What is the total kinetic energy of the wheel?

a. 9.0 J

b. 18 J

c. 27 J

d. 36 J

e. 54 J

17. A 60.0-kg skater begins a spin with an angular speed of 6.0 rad/s. By changing the position of her arms, the skater decreases her moment of inertia by 50%. What is the skater's final angular speed?

a. 3.0 rad/s

b. 4.5 rad/s

c. 9.0 rad/s

d. 12 rad/s

e. 18 rad/s

18. Gina's favorite exercise equipment at the gym consists of various springs. In one exercise, she pulls a handle grip attached to the free end of a spring to 0.80 m from the unstrained position. The other end of the spring (spring constant = 53 N/m) is held in place by the equipment frame. What is the magnitude of the force that Gina is applying to the handle grip?

a. 31 N

b. 36 N

c. 42 N

d. 54 N

e. 66 N

19. Which one of the following statements is true concerning an object executing simple harmonic motion?

a. Its velocity is never zero.

b. Its acceleration is never zero.

c. Its velocity and acceleration are simultaneously zero.

d. Its velocity is zero when its acceleration is a maximum.

e. Its maximum acceleration is equal to its maximum velocity.

20. The position of a simple harmonic oscillator is given by x(t) = (0.5m)cos(πt/3) where t is in seconds. What is the period of the oscillator?

a. 0.17 s

b. 0.67 s

c. 1.5 s

d. 3.0 s

e. 6.0 s

21. A spring required a force of 1.0 N to compress it 0.1 m. How much work is required to stretch the spring to 0.4 m?

a. 0.4 J

b. 0.6 J

c. 0.8 J

d. 2 J

e. 4 J

22. In a certain clock, a pendulum of length L1 has a period T1 = 0.95 s. The length of the pendulum is adjusted to a new value L2 such that T2 = 1.0 s. What is the ratio L2/L1?

a. 0.90

b. 0.95

c. 1.0

d. 1.1

e. 1.3

23. Complete the following sentence: Resonance occurs in harmonic motion when

a. the system is overdamped.

b. The system is critically damped.

c. The energy in the system is a minimum.

d. The driving frequency is the same as the natural frequency of the system.

e. The energy in the system is proportional to the square of the motion's amplitude.

24. A cable stretches by an amount d when it supports a crate of mass M. The cable is cut in half. What is the mass of the load that can be supported by either half of the cable if the cable stretches by an amount d?

a. M/4

b. M/2

c. M

d. 2M

e. 4M

25. A plastic box has an initial volume of 2.00 m3. It is then submerged below the surface of a liquid and its volume decreases to 1.96 m3. What is the volume strain on the box?

a. 0.02

b. 0.04

c. 0.2

d. 0.4

e. 0.98