Week 3: Rotational Dynamics

Michael Kurniawan 1910403011 University of Cincinnati ROTATIONAL DYNAMICS 1. A disk with mass 𝑚 and radius 𝑅 is rotating with an angular velocity. Calculate the rotational kinetic energy of the disk. 2. A homogeneous rod with length 𝐿 and massn𝑀 is rotating about an axis through one of its ends with an angular velocity 𝜔. Calculate the moment of inertia of the rod and its rotational kinetic energy 3. A flywheel with a moment of inertia 𝐼 starts from rest and is then accelerated with a constant torque 𝜏 for 𝑡 seconds. Determine the final angular velocity of the flywheel 4. A disk with radius 𝑅 and mass 𝑚 is released from rest at the top of an inclined plane with an angle of inclination 𝜃. Calculate the angular acceleration of the disk as it rolls without slipping. When a disk rolls without slipping, the linear acceleration aaa of its center of mass is related to the angular acceleration α by: The forces acting on the disk are gravity, normal force, and friction. The component of gravitational force along the plane is 𝑚𝑔sin𝜃, and the torque due to friction 𝑓 is 𝑓𝑅. Using Newton's second law for rotation: Using Newton's second law for linear motion along the incline: 5. A solid cylinder with radius 𝑅 and mass 𝑀 is rotating with an angular velocity 𝜔. Calculate the centripetal force acting on a small particle on the surface of the cylinder. The centripetal force 𝐹𝑐 acting on a small particle of mass 𝑚 on the surface of the cylinder is given by: For a small particle on the surface of a solid cylinder, assume it has a mass m. The centripetal force is: If the entire mass M of the cylinder were considered to be concentrated at the surface (which isn't typically the case, but if we consider a small surface particle, this is reasonable), then: 6. A thin rod of length 𝐿 and mass 𝑀 rotates with an angular velocity 𝜔 about an axis through its center and perpendicular to its length. Calculate the moment of inertia of the rod. The moment of inertia I of a thin rod rotating about an axis through its center and perpendicular to its length is: 7. A sphere with mass 𝑚 and radius 𝑟 rotates with an angular velocity 𝜔. If the sphere experiences friction that generates a torque 𝜏, calculate the time it takes for the sphere to come to a stop. The angular deceleration α caused by the torque τ is: The moment of inertia I of a solid sphere is: The time t it takes for the sphere to stop is given by: 8. Two solid wheels with the same mass 𝑚 but radii 𝑅1 and 𝑅2 are connected by a belt that does not slip. If one wheel is rotated with an angular velocity 𝜔, calculate the angular velocity of the other wheel. Since the belt does not slip, the linear velocities at the rims of both wheels must be equal. For the first wheel: 9. A disk with mass 𝑀 and radius 𝑅 is rotating with an angular velocity 𝜔. Suddenly, a small object with mass 𝑚 is placed on the edge of the disk. Calculate the new angular velocity of the disk after the object is placed. When the small object is placed on the edge, the new moment of inertia If is: 10. A torsional pendulum consists of a homogeneous rod with length 𝐿 and mass 𝑀 mounted in the middle so it can rotate freely. Calculate the period of the torsional pendulum if the torsional spring constant is 𝑘.