NYJC EJC 2026 Rotational Motion Tutorial 2

Page 1 Tutorial 2 Rotational Energy and Rotational Dynamics Rotational Energy 1 A tall, cylindrical chimney falls over when its base is ruptured. Treat the chimney as a thin rod of length 55.0 m. At the instant it makes an angle of 35.0° with the vertical as it falls, what are (a) the radial acceleration of the top, and (b) the tangential acceleration of the top. (Hint: Use energy considerations, not a torque.) (c) At what angle θ is the tangential acceleration equal to g? 2 A uniform spherical shell of mass M = 4.5 kg and radius R = 8.5 cm can rotate about a vertical axis on frictionless bearings. A massless cord passes around the equator of the shell, over a pulley of rotational inertia I = 3.0 × 10-3 kg m2 and radius r = 5.0 cm, and is attached to a small object of mass m = 0.60 kg. There is no friction on the pulley’s axle; the cord does not slip on the pulley. Using energy consideration, determine the speed of the object when it has fallen 82 cm after being released from rest. 3 In the figure a wheel of radius 0.20 m is mounted on a frictionless horizontal axis. The rotational inertia of the wheel about the axis is 0.40 kg m 2 . A massless cord wrapped around the wheel’s circumference is attached to a 6.0 kg box. The system is released from rest. When the box has a kinetic energy of 6.0 J, what are (a) the wheel’s rotational kinetic energy and (b) the distance the box has fallen? 4 A uniform cylinder of radius 10 cm and mass 20 kg is mounted so as to rotate freely about a horizontal axis that is parallel to and 5.0 cm from the central longitudinal axis of the cylinder. (a) What is the rotational inertia of the cylinder about the axis of rotation? (b) If the cylinder is released from rest with its central longitudinal axis at the same height as the axis about which the cylinder rotates, what is the angular speed of the cylinder as it passes through its lowest position?

Page 2 Rotational Dynamics 5 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 angle θ = 20° with the horizontal. The box accelerates down the surface at 2.0 m s-2. What is the rotational inertia of the wheel about the axle? 6 A wheel of radius 0.20 m is mounted on a frictionless horizontal axis. The rotational inertia of the wheel about the axis is 0.050 kg m2. A massless cord wrapped around the wheel is attached to a 2.0 kg block that slides on a horizontal frictionless surface. If a horizontal force of magnitude P = 3.0 N is applied to the block as shown in the figure, what is the magnitude of the angular acceleration of the wheel? State any assumptions that you made in your calculation. 7 In Fig. (a), an irregularly shaped plastic plate with uniform thickness and density (mass per unit volume) is to be rotated around an axle that is perpendicular to the plate face and through point O. The rotational inertia of the plate about that axle is measured with the following method. A circular disk of mass 0.500 kg and radius 2.00 cm is glued to the plate, with its center aligned with point O Fig. (b). A string is wrapped around the edge of the disk the way a string is wrapped around a top. Then the string is pulled for 5.00 s. As a result, the disk and plate are rotated by a constant force of 0.400 N that is applied by the string tangentially to the edge of the disk. The resulting angular speed is 114 rad/s. What is the rotational inertia of the plate about the axle? 8 The figure shows particles 1 and 2, each of mass m, fixed to the ends of a rigid massless rod of length L1 + L2, with L1 = 20 cm and L2 = 80 cm. The rod is held horizontally on the fulcrum and then released. What are the magnitudes of the initial accelerations of (a) Particle 1, and (b) Particle 2?

Page 3 Answers 1 (a) 5.32 m s-2 (b) 8.44 m s-2 (c) 41.8° 2 1.4 m s-1 3 (a) 10 J (b) 0.27 m 4 (a) 0.15 kg m2 (b) 11 rad s-1 5 0.054 kg m2 6 4.63 rad s-2 7 2.51 × 10-4 kg m2 8 (a) 1.7 m s-2 (b) 6.9 m s-2

Page 4 Worked Solution 1

Page 5 2 The rotational inertia of the spherical shell is 2MR2/3, so th kinetic energy (after the object descended distance h is 3 THINK As the box falls, gravitational force gives rise to a torque that causes the wheel to rotate. EXPRESS We employ energy methods to solve this problem; thus, considerations of positive versus negative sense (regarding the rotation of the wheel) are not relevant.

Page 6 4 5 We choose positive coridnate direction so that each is accelerating positively,

Page 7 6 THINK The applied force P accelerates the block. In addition, it gives rise to a torque that causes the wheel to undergo angular acceleration. 7 Combining 𝐾 = 1 2 𝐼𝜔2 and 𝜏𝑛𝑒𝑡 = 𝐼𝛼 we have 𝑅𝐹 = 𝐼𝛼, where 𝛼 is given by 𝜔 𝑡 (since 𝜔0 = 0). We also use the fact that

Page 8 8 With counterclockwise positive, the angular acceleration 𝛼 for both masses satisfies