A.4 剛體力學（HL）：練習題

A force of \(40\text{ N}\) is applied perpendicularly to the end of a lever that is \(0.50\text{ m}\) long. What is the magnitude of the torque produced about the pivot at the other end of the lever?

A constant net torque of \(12\text{ N m}\) is applied to a rigid body with a moment of inertia of \(4.0\text{ kg m}^2\). If the body starts from rest, what is its angular velocity after \(5.0\text{ seconds}\)?

A horizontal disc of moment of inertia \(I_d\) is rotating freely about a vertical axis through its center with angular velocity \(\omega_0\). A small piece of clay of mass \(m\) is dropped vertically onto the disc and sticks to it at a distance \(r\) from the axis. What is the final angular velocity of the system?

Two point masses, each of mass \(m\), are fixed to the ends of a light rod of length \(L\). The system rotates about an axis perpendicular to the rod passing through its center. What is the moment of inertia of this system?

A solid sphere and a thin hoop have the same mass \(M\) and radius \(R\). Both are rotating about an axis passing through their center. Which statement correctly compares their moment of inertia \(I\)?

A mechanic applies a force of \(22\text{ N}\) to the end of a wrench that is \(0.25\text{ m}\) long. If the force is applied at an angle of \(45^\circ\) to the wrench, calculate the magnitude of the torque applied to the bolt.

A uniform rod of length \(L\) and mass \(M\) rotates about an axis through its center. Explain how its moment of inertia changes if the axis is moved to one of its ends.

A uniform solid sphere (moment of inertia \(I = \frac{2}{5}MR^2\)) rolls without slipping from rest down an incline of vertical height \(h\). Calculate the ratio of its final translational velocity to the velocity it would reach if it slid down a frictionless incline of the same height, and explain the difference using energy principles.

A small 0.20 kg block is placed on a horizontal turntable at a distance of 0.40 m from the center. The turntable is rotating at 2.0 rad s^-1. A mechanism then pulls the block slowly toward the center until it is at a distance of 0.15 m from the center. Assume the turntable's own moment of inertia is negligible compared to the block.
(a) State the principle that allows you to find the new angular velocity of the block.
(b) Calculate the initial and final moments of inertia of the block.
(c) Determine the final angular velocity of the block.
(d) Calculate the work done by the mechanism to pull the block toward the center.

A uniform solid cylinder of mass \(M\) and radius \(R\) is released from rest at the top of an incline of height \(h\) and angle \(\theta\). The cylinder rolls without slipping. The moment of inertia of a solid cylinder is \(I = \frac{1}{2}MR^2\).
(a) By considering the conservation of energy, derive an expression for the linear speed \(v\) of the cylinder's center of mass when it reaches the bottom of the incline.
(b) Determine the magnitude of the static frictional force acting on the cylinder in terms of \(M\), \(g\), and \(\theta\).
(c) If a thin hollow hoop of the same mass and radius were used instead, explain without further calculation whether its final speed would be greater, smaller, or the same as the cylinder.