Name Period Date Rotation: Worksheet 9 Newton’s Second Law for Rotation 1. The length of a bicycle pedal arm is 0.152 m , and a downward force of 111 N is applied by the foot. Calculate the magnitude of the torque about the pivot point when the arm makes the following angle with the vertical. a. 30. b. 90. c. 180 2. A merry-go-round has a rotational inertia of 3000. kg.m2. A clockwise 2500 Nm torque is applied to a pulley connected to the bottom of the ride. There is a 500. Nm frictional torque in the axle. Determine the angular acceleration for the merry-goround. 3. A wheel of radius 0.20 m is mounted on a frictionless horizontal axle. A massless cord is wrapped around the wheel and attached to a 2.0 kg box that slides on a frictionless surface inclined at 20. with the horizontal. The box accelerates down the incline at 2.0 m 2 . s a. Calculate the tension in the string. Draw a force diagram of the box. b. What is the tangential component of the linear acceleration of the outside of the wheel? c. What is the angular acceleration of the wheel? d. What is the rotational inertia of the wheel about the axle? Rotation: Worksheet 9 page 2 4. When a torque of 32.0 Nm is applied to a certain wheel of mass 3.25 kg , it acquires an angular acceleration of 25.0 rad 2 . s a. What is the b. If the wheel can be c. What would the rotational inertia approximated as a wheel’s radius be if it of the wheel? hoop, what is its must be approximated radius? as a disk? 5. A pottery wheel, initially at rest, is assumed to be a uniform disk of mass 3.0 kg and radius 20.0 cm. A 25.0 N force is applied tangentially to the rim. a. Determine the b. If the force is applied c. What will be the for 30.0 s, find the angular angular displacement acceleration of this for this wheel from instantaneous angular t = 15 s until t = 30 s? pottery wheel. velocity at 30.0 s. 2 6. A thin spherical shell I MR 2 has a radius of 1.90 m . An applied torque of 3 960 Nm imparts to the shell an angular acceleration equal to 6.20 rad 2 about an s axis through the center of the shell. a. What is the rotational inertia of b. Calculate the mass of the shell. the shell about the axis of rotation? Rotation: Worksheet 9 7. Two blocks, each of mass m , are suspended from the ends of a rigid, weightless rod of length l1 l 2 , with l1 20. cm and l 2 80. cm . The rod is held in the horizontal position shown and then released. page 3 l1 l2 m m a. Draw a force diagram of the rod in the position shown. b. Find an expression for the rotational inertia of the system in terms of m, l1 , and l 2 . (Treat the two masses as point particles located at the ends of the rod.) c. Calculate the angular acceleration of the rod the instant it is released. d. Calculate the linear acceleration of each block the instant they are released. Rotation: Worksheet 9 F page 4 L d d / 2 8. A constant force F is applied to the rightmost ball of a two ball object, each of mass m, connected by a massless rod. The force is applied in such a way that it always acts perpendicular to the rod as the object rotates in the horizontal plane from rest with no friction. Express your answers in terms of M, L, t and F only. a. What is the b. What torque acts on the c. What is the resulting rotational inertia of object? angular acceleration of the object? the object? The object rotates for a time t. c. What is the angular velocity of the rightmost ball? d. What is the angular velocity of the leftmost ball? f. What is the radial acceleration of each ball? e. What is the linear velocity of each ball? g. What is the tangential acceleration of each ball? Rotation: Worksheet 9 page 5 M R m 9. A box of mass m is hung from a massless rope which is tied around a pulley that can be approximated as a hoop of mass M and radius R as in the diagram above. The rope does not slip. a. Draw force b. Compare the c. Write Newton’s 2nd Law diagrams of the magnitudes of equations for the pulley and the box in terms of m, M, R, FT, g, hoop and the the forces acting and a. You must indicate box. on the box. Justify your directions (+ or -) for all vector answer. quantities using a conventional coordinate system. Answer the following in terms of m, M, and g only. a. What is the acceleration of the box? b. What is the force of tension in the rope?