# Rotational axes and maximum stresses

See the attached file.

1. Two solid aluminum shafts (G = 3.7x106 psi) are connected by the gears shown. Ends B and D are fixed. What is the angle through which end C rotates?

2. For the beam of the cross section shown, determine the location of the neutral axis from the bottom of the beam.

3. For the beam of question 8, determine the value of the second moment of area about the neutral axis.

4. Two vertical forces of 3 kN each are applied to the beam of questions 8 and 9. Determine the maximum stress in section AB if L = 0.8 m.

5. The 6 x 12 in timber beam has been strengthened by bolting to it the 5 x ½ in steel reinforcement shown. The modulus of elasticity for wood is 1.8 x 106 psi and for steel is 29 x 106 psi. The beam is bent about a horizontal axis by a couple M = 450 kip-in. What is the maximum stress in the wood?

6. The 6 x 12 in timber beam has been strengthened by bolting to it the 5 x ½ in steel reinforcement shown. The modulus of elasticity for wood is 1.8 x 106 psi and for steel is 29 x 106 psi. The beam is bent about a horizontal axis by a couple M = 450 kip-in. What is the maximum stress in the steel?

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#### Solution Summary

Six problems are answered having to do with maximum stresses and rotational axes. Thorough calculations are provided.

25 Multiple Choice question all mechanics, Fully explained

1. A bicycle travels 141 m along a circular track of radius 15m. 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. What is the angular speed in rad/s of the second hand of a watch?

a. 1.7 à? 10-3 rad/s

b. 0.10 rad/s

c. 0.02 rad/s

d. 6.28 rad/s

e. 60 rad/s

3. During the spin-dry cycle of a washing machine, the motor slows from 95 rad/s to 30 rad/s while the turning the drum through an angle of 402 radians. What is the magnitude of the angular acceleration of the motor?

a. 64 rad/s2

b. 32 rad/s2

c. 10 rad/s2

d. 20 rad/s2

e. 1.0 rad/s2

4. During the time a compact disc (CD) accelerates from rest to a constant rotational

speed of 477 rev/min, it rotates through an angular displacement of 0.250 rev. What is the angular acceleration of the CD?

a. 126 rad/s2

b. 358 rad/s2

c. 794 rad/s2

d. 866 rad/s2

e. 901 rad/s2

5. A grindstone of radius 4.0 m is initially spinning with an angular speed of 8.0 rad/s. The angular speed is then increased to 10 rad/s over the next 4.0 seconds. Assume that the angular acceleration is constant. What is the average angular speed of the grindstone?

a. 0.5 rad/s

b. 2.0 rad/s

c. 4.5 rad/s

d. 9.0 rad/s

e. 18 rad/s

6. A roulette wheel with a 1.0-m radius reaches a maximum angular speed of 18 rad/s before it stops 35 revolutions (220 rad) after attaining the maximum speed. How long did it take the wheel to stop?

a. 12 s

b. 48 s

c. 3.7 s

d. 8.8 s

e. 24 s

7. What is the tangential speed of a lug nut on a wheel of a car if the lug nut is located 0.114 m from the axis of rotation and the wheel is rotating at 6.53 rev/sec?

a. 0.745 m/s

b. 1.49 m/s

c. 2.98 m/s

d. 4.68 m/s

e. 9.36 m/s

8. A rigid body rotates about a fixed axis with a constant angular acceleration. Which one of the following statements is true concerning the tangential acceleration of any point on the body?

a. Its magnitude is zero m/s2

b. It depends on the angular velocity

c. It is equal to the centripetal acceleration

d. It is constant in both magnitude and direction

e. It depends on the change in the angular velocity

9. A circular hoop rolls without slipping on a flat horizontal surface. Which one of the following is necessarily true?

a. All points on the rim of the hoop have the same speed

b. All points on the rim of the hoop have the same velocity

c. Every point on the rim of the wheel has a different velocity

d. All points on the rim of the hoop have acceleration vectors that are tangent to the hoop

e. All points on the rim of the hoop have acceleration vectors that point toward the center of the hoop

10. A wrench is used to tighten a nut as shown in the figure above. A 12-N force is applied 7.0 cm from the axis of rotation. What is the torque due to the applied force?

a. 0.58 N ? m

b. 0.84 N ? m

c. 1.71 N ? m

d. 14 N ? m

e. 58 N ? m

11. Complete the following statements: A body is in translational equilibrium

a. Only if it is at rest

b. Only if it is moving with constant velocity

c. Only if it is moving with constant velocity

d. If it is either at rest or moving with constant velocity

e. If it is moving with either constant velocity or constant acceleration

12. What, approximately, is the magnitude of the downward force exerted on the fulcrum?

a. Zero newtons

b. 100 N

c. 600 N

d. 800 N

e. 1000 N

13. Consider four point masses located as shown in the sketch. The acceleration due to gravity is the same everywhere. What is the x coordinate of the center of gravity for this system?

a. 2.0 m

b. 2.7 m

c. 3.0 m

d. 3.3 m

e. 3.8 m

14. Two uniform solid spheres, A and B have the same mass. The radius of sphere B is twice that of sphere A. The axis of rotation passes through each sphere. Which one of the following statements concerning the moments of inertia of these spheres is true?

a. The moment of inertia of A is one-fourth that of B

b. The moment of inertia of A is one-half that of B

c. The moment of inertia of A is 5/4 that of B

d. The moment of inertia of A is 5/8 that of B

e. The two spheres have equal moments of inertia

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 50-kg rider on a moped of mass 75 kg is traveling with a speed of 20 m/s. Each of the two wheels of the moped has a radius of 0.2 m and a moment of inertia of 0.2 kg ? m2. What is the total rotational kinetic energy of the wheels?

a. 80 J

b. 100 J

c. 500 J

d. 2000 J

e. 4000 J

17. A solid sphere of radius R rotates about a diameter with an angular speed ω. The sphere then collapses under the action of internal forces to a final radius R/2. What is the final angular speed of the sphere?

a. ω/4

b. ω/2

c. ω

d. 2ω

e. 4ω

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 its 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 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. A ball hung from a vertical spring oscillates in simple harmonic motion with an angular frequency of 2.6 rad/s and an amplitude of 0.075 m. What is the maximum acceleration of the ball?

a. 0.13 m/s2

b. 0.20 m/s2

c. 0.51 m/s2

d. 2.6 m/s2

e. 35 m/s2

20. A ping-pong ball weighs 2.5 à? 10-2N. The ball is placed inside a cup that sits on top of a vertical spring. If the spring is compressed 0.055 m and released, the maximum height above the compressed position that the ball reaches is 2.84 m. Neglect air resistance and determine the spring constant.

a. 47 N/m

b. 24 N/m

c. 11 N/m

d. 5.2 N/m

e. 2.6 N/m

21. What is the period of a pendulum consisting of a 6-kg object oscillating on a 4-m string?

a. 0.25 s

b. 0.50 s

c. 1.0 s

d. 2.0 s

e. 4.0 s

22. Which one of the following terms is used to describe a system in which the degree of damping is just enough to stop the system from oscillating?

a. Slightly damped

b. Under-damped

c. Critically damped

d. Over-damped

e. Resonance

23. A simple harmonic oscillator with a period of 2.0 s is subject to damping so that it loses one percent of its amplitude per cycle. About how much energy does this oscillator lose per cycle?

a. 0.5 %

b. 1.0 %

c. 1.4 %

d. 2.0 %

e. 4.0 %

24. A cable stretches by an amount d when it supports a crate of mass M. The cable is replaced by another cable of the same material having the same length but twice the diameter. What is the mass of the load that can be supported by the thicker cable if it stretches by an amount d?

a. M/4

b. M/2

c. M

d. 2M

e. 4M

25. Which one of the following statements concerning Hook's law is false?

a. Hook's law relates stress and strain.

b. Hook's law is valid only for springs.

c. Hook's haw can be verified experimentally.

d. Hook's law can be applied to a wide range of materials.

e. Hook's law is valid only within the elastic limit of a given material.