A uniform rigid rod of mass \(M\) and length \(L\) is suspended vertically from a frictionless pivot at its top end. The rotational inertia of the rod about the pivot is \(I = \dfrac{1}{3}ML^2\). A small sphere of mass \(m\) is launched horizontally with initial speed \(v_0\) toward the rod. The sphere strikes the very bottom of the rod and sticks to it. The collision occurs so quickly that the rod does not noticeably move from its vertical position until after the collision is over. The sphere can be treated as a point mass.

A uniform disk of mass \(M\) and radius \(R\) is mounted on a horizontal, frictionless axle that passes through a point \(P\) located a distance \(\dfrac{R}{2}\) from the center of the disk. The disk is held in a position such that the center of the disk and the axle are at the same horizontal level, as shown in the figure. At the moment the disk is released from rest, which of the following correctly describes the torque about the axle and the magnitude of the upward vertical force \(F_{axle}\) exerted by the axle on the disk?

A block of mass \(M\) and a solid sphere of mass \(M\) are released from rest at the top of two identical ramps of height \(h\) and angle \(\theta\). The ramp for the block is frictionless, while the ramp for the sphere has sufficient friction for the sphere to roll without slipping. Which of the following graphs best represents the speed \(v\) of the center of mass for the block and the sphere as a function of time \(t\) while they are on the ramps?

A small ball of mass \(m\) slides across a horizontal, frictionless surface with a velocity \(v\) toward a uniform rod of mass \(M\) and length \(L\). The rod is initially at rest and is free to rotate about a frictionless pivot at one of its ends. The ball strikes the rod at its midpoint and sticks to it. Which of the following correctly states whether the angular momentum of the ball-rod system about the pivot is conserved during the collision and provides a correct justification?

A figure skater with an initial rotational inertia \(I_0\) is spinning on ice with an initial angular velocity \(\omega_0\). Friction between the skater’s blades and the ice is negligible. The skater pulls their arms and one leg inward toward their axis of rotation, which decreases their rotational inertia to a final value \(I_f\). Let \(K_0\) represent the initial rotational kinetic energy of the skater. Which of the following expressions correctly represents the skater’s final rotational kinetic energy \(K_f\)?

Two uniform solid spheres, Sphere 1 and Sphere 2, are released from rest at the top of an incline that makes an angle \(\theta\) with the horizontal. Sphere 1 has mass \(M_1\) and Sphere 2 has mass \(M_2\), where \(M_1 = 2M_2\). Both spheres have the same radius \(R\) and roll down the incline without slipping. What is the ratio \(a_1/a_2\) of the linear acceleration of the center of mass of Sphere 1 to that of Sphere 2?

An engineer is testing two different designs for a rotating flywheel. Design A consists of a uniform solid disk with mass \(M\) and radius \(R\) mounted on a low-friction axle. When a motor applies a constant net torque \(\tau\) to Design A, it experiences an angular acceleration \(\alpha_0\). Design B consists of a uniform solid disk with mass \(2M\) and radius \(2R\). If the motor is adjusted to apply a constant net torque of \(4\tau\) to Design B, what is the resulting angular acceleration of Design B in terms of \(\alpha_0\)?

A spring of ideal spring constant \(k\) hangs vertically from a ceiling. When the spring is unextended, its bottom end is at position \(y = 0\). The positive \(y\)-direction is defined as downward. A block of mass \(m\) is attached to the spring and gently lowered until it hangs at rest at its equilibrium position \(y_{eq}\). The block is then pulled down an additional distance \(A\) to a maximum position \(y_{max} = y_{eq} + A\) and released from rest.

A block of mass \(M\) is attached to an ideal spring and undergoes simple harmonic motion on a frictionless horizontal surface. The equilibrium position of the block is at \(x = 0\). Which of the following graphs best represents the acceleration \(a\) of the block as a function of its displacement \(x\)?

A block of mass \(M\) is attached to the lower end of a vertical spring with spring constant \(k\), while the upper end of the spring is fixed to a ceiling. A second block of mass \(m\) is placed on top of the first block. The two-block system is set into vertical simple harmonic motion with amplitude \(A\). At the instant the blocks pass through the equilibrium position while moving downward, which of the following correctly describes the magnitude of the force exerted by the spring on the bottom block, \(F_{spring}\), and the magnitude of the normal force exerted by the bottom block on the top block, \(F_{normal}\)?