Honors Physics Assignment Rotational Mechanics Reading Chapters 10 and 11 Objectives/HW The student will be able to: HW: 1 Determine the torque of an applied force and solve related problems. (t = rx r r 1 11 F) 2 Determine the angular acceleration achieved by application of torque 12 17 (τ=iα) 3 Determine the moment of inertia for a system of masses or solid body 18 25 and solve related problems. I~MR 2 4 Conceptual questions. 25 31 Homework Problems τ=f r 1. What is the maximum torque exerted by a 55-kg person riding a bike if the rider puts all her weight on each pedal when climbing a hill? The pedals rotate in a circle of radius 17cm. 2. A person exerts a force of 28N on the end of a door 84 cm wide. What is the magnitude of the torque if the force is exerted (a) perpendicular to the door, and (b) at a 60 angle to the face of the door? 3. Calculate the net torque about the axle of the wheel shown 4. The bolts on the cylinder head of an engine require tightening to a torque of 80m N. (a) If a wrench is 30cm long, what force perpendicular to the wrench must the mechanic exert at its end? (b) If the six-sided bolt head is 15mm in diameter, estimate the force applied near each of the six points by a socket wrench.
5. If the coefficient of static friction between tires and pavement is 0.75, calculate the minimum torque that must be applied to the 66cm diameter tire of a 1250-kg automobile in order to lay rubber (make the wheels to spin, slipping as the car accelerates). Assume each wheel supports an equal share of the weight. 6. As shown below a square metal plate with sides of length 20.0 cm is subject to three forces: F 1 = 60.0 N, 180.0, F 2 = 90.0 N, 30.0, F 3 = 100.0 N, 180.0. (a) Find the torque of each force about the lower left corner of the square. (b) Find the torque of each force about the midpoint of the upper side of the square. (c) What additional horizontal force applied at the lower left corner would make the net torque equal zero about this midpoint? 7. The pedal of a bicycle has a radius of 0.17 m relative to the center of the crank. A rider of mass 80.0 kg puts all of his weight on the pedal as it moves from top to bottom. Find the torque on the crank when the crank arm is in each of the following positions: (a) r = 0.17 m, 90.0, (b) r = 0.17 m, 45.0, (c) r = 0.17 m, 0.0. 8. The nut on the drive axle of a 1974 VW bus must be torqued to 407 Nm. In order to apply this much torque the mechanic attaches a wrench to the nut and places a cheater pipe over the wrench in order to lengthen it. (a) If length from the center of the nut to the end of the cheater is 0.500 m, what is the minimum force that must be applied? (b) If the mechanic is only capable of exerting 445 N of force how long must the cheater be? (c) Draw a picture showing the most effective way to arrange the cheater and apply the force. 9. In order to stand on your toes (on the ball of your foot, usually), your Achilles tendon must pull up on your calcaneus (heelbone). For an 80kg person, how much tension must be in the Achilles tendon to stand on the ball of your foot? 4cm 16cm 10. A 1250 kg car is stopped at a red light on a hill of incline 10.0. (a) Assuming the weight is equally distributed on the four tires of radius 30.5 cm each, what is the torque due to friction about the center of each wheel? (b) This torque must be countered by the brakes to prevent the car rolling. If the car has disk brakes and the calipers are located 11.4 cm from the center of the wheel, what is the amount of tangential force on the calipers?
11. This old sign is barely hanging on, but it s still up for now. The sign itself weighs 100N. It is hanging 2m out from the wall. The cable holding it up is connected to the end of the 3.5m long rod. It is at a 28 angle to the rod. The rod itself weighs 25N. (a) How much tension is there in the cable? (b) In order for there to be no vertical force needed where the rod rests against the wall, where should the sign be hung along the rod? τ=iα 12. A uniform cylindrical grinding wheel of mass 5.00 kg and radius 20.0 cm is turned by an electric motor. Once it is turned on it has an angular acceleration of 5.0 rad/s 2. Ignore friction in the bearings of the wheel. (a) Find the amount of torque exerted on the wheel by the motor. (b) The motor is switched off and a piece of metal is pressed against the edge of the wheel with a force of 45.0 N. If the coefficient of friction is 0.500 what is the angular acceleration to slow the wheel down? 13. An Atwood s machine is formed by a string passing over a solid pulley of mass 50.0 grams and diameter 5.00 cm. The masses on each end of the string are: m 1 = 200.0 g, and m 2 = 300.0 g. Friction is negligible. (a) Find the linear acceleration of m 1 as one mass falls and the other rises. (b) Find the angular acceleration of the pulley. (c) What would be the linear acceleration of m 1 if the pulley were massless? 14. A teeter-totter (or seesaw ) of length 3.00 m and mass 50.0 kg is pivoted at its center. A boy of mass 40.0 kg sits 1.40 m left of center and a girl of mass 30.0 kg sits 1.30 m right of center. Ignore friction in the bearing. For the instant when the teeter-totter is horizontal find the angular acceleration. 15. A day-care worker pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it at 0.3 rad/s 2. Assume the merry-go-round is a disk of radius 2.5m and has a mass of 800kg, and two children (each with mass of 25kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. What force is required?
16. The forearm accelerates a 3.6kg ball at 7.0 m/s 2 by means of the triceps muscle, as shown. Calculate (a) the torque needed, and (b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm. 17. In order to get a flat uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets. If the satellite has a mass of 2000kg and a radius of 3.0m, what is the required steady force of each rocket if the satellite is to have an angular acceleration of.63rad/s 2? MOI 18. Calculate the moment of inertia of a 14.0 kg sphere of radius 0.623m when the axis of rotation is through its center. 19. Calculate the moment of inertia of a 66.7 cm diameter bicycle wheel. The rim and tire have a combined mass of 1.25kg. The mass of the hub can be ignored. Why? 20. Historically, bicycle wheels have a diameter of 26 inches. Many newer mountain bikes have a wheel diameter of 29 inches, cleverly called 29 ers. The idea is that by having bigger wheels, it s easier to roll over rocks and roots. However, it also increases the MOI by placing the mass further from the hub. By what factor does the MOI increase in going from 26 to 29 wheels? (You can assume all the mass is at the edge, but you should not assume that the mass is the same in both 26 and 29 tires.) 21. An oxygen molecule consists of two oxygen atoms whose total mass is 5.3x10-26 kg, and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9x10-46 kg m 2. Estimate, from these data, the effective distance between the atoms. 22. A small 1.05kg ball on the end of a light rod is rotated in a horizontal circle of radius 0.900m. Calculate (a) the moment of inertia of the system about the axis of rotation, and the (b) torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.08000N on the ball. 23. In a physics demonstration a student stands upright, centered on a platform that can rotate freely. The student holds in each hand a 0.500 kg mass. Find the moment of inertia of the two masses about the axis of rotation when (a) the two masses are held at the shoulders and are 0.50 m apart, and (b) the two masses are held at arms length and are 1.50 m apart.
24. Compact masses of 4.00 kg and 5.00 kg are attached to the ends of a rod of length 90.0 cm and negligible mass. (a) Find the moment of inertia about the center of the rod. (b) Find the moment of inertia about each end of the rod. (c) Find the moment of inertia about the center of mass. Assume the axis is perpendicular to the rod in each case. 25. Use reference Table 8.1. (a) Find the moment of inertia of a solid disk of mass 3.00 kg and radius 0.500 m about an axis passing perpendicularly through its center. (b) Repeat for a hoop of the same mass and radius. (c) Repeat for a solid sphere of the same mass and radius. 26. Mammals that depend on being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body. Compare a buffalo to a hippo, or a bear and a gazelle. On the basis of rotational dynamics, explain why this distribution of mass is advantageous. 27. Explain the physics justification for choking up on a baseball bat. 28. If there were a great migration of people toward the equator, how would this affect the length of a day? 29. There are 2 meter sticks with clamps on them. One has the masses at the ends, the other has the masses near the middle. Hold one in each hand and try to twist them back and forth. What do you observe? Explain. 30. Imagine a person carrying a heavy luggage bag in one hand. Why does this person stick their other arm straight out? 31. I have always wanted to be a tightrope walker, but never found the idea of falling to my death a pleasant thought. So, I hold on to a heavy, long pole while I do my routine. I can hold the pole horizontally, so as I walk, there is a lot of pole sticking out to each side of me. Explain why this is helpful (as opposed to no pole).
Answers: 1. 94 Nm 2. a. 24 Nm b. 20 Nm 3. 1.50 Nm CW 4. a. 270 N b. 1800 N 5. 770 Nm 6. a. t 1 =12.0 Nm CCW t 2 = 6.59 Nm CW t 3 = 0 b. t 1 = 0 t 2 = 4.50 Nm CCW t 3 = 20.0 Nm CW c. 77.5 N, 0 7. a. 0 b. 96 Nm CW c. 140 Nm CW 8. a. 814 N b. 0.915 m c. 9. 3200 N 10. a. 166 Nm CCW b. 1450 N 11. 12. a. 0.50 Nm b. -45 rad /s 2 13. a. 1.87 m/s 2 b. 74.7 rad/s 2 c. 1.96 m/s 2 14. 1.00 rad/s 2 15. 16. a. 7.6 Nm CCW b. 300 N 17. 470 N 18. 2.17 kgm 2 19. 0.139 kgm 2 20. 1.39 21. 1.2 x 10-10 m 22. a. 0.851 kgm 2 b. 0.0720 Nm 23. a. 0.063 kgm 2 b. 0.563 kgm 2