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 an ideal horizontal spring with spring constant \(k\). The system oscillates on a frictionless surface with amplitude \(A\). Which of the following expressions represents the kinetic energy of the block when its displacement from the equilibrium position is \(x = \dfrac{1}{3}A\)?

Two uniform, thin rods, Rod A and Rod B, both have the same mass \(M\). Rod A has length \(L\) and Rod B has length \(2L\). Both rods are pivoted at one end so they can rotate in a vertical plane. The rods are held in a horizontal position and released from rest at the same time. What is the ratio \(\dfrac{\alpha_B}{\alpha_A}\) of the magnitude of the initial angular acceleration of Rod B to that of Rod A?

A drone of mass \( m \) is flying in a straight horizontal line at a constant speed \( v \) and a constant altitude \( H \) above the ground. A stationary observer is located at point \( P \) on the ground. At a certain moment, the horizontal distance between the drone and the observer is \( x \). Which of the following is a correct expression for the magnitude of the angular momentum of the drone relative to the observer at this moment?

A uniform rigid rod of mass \(M\) and length \(D\) is suspended vertically from a horizontal, frictionless pivot at its top end. The rotational inertia of the rod about the pivot is \(I_{rod} = \dfrac{1}{3}MD^2\). A small lump of clay of mass \(m\) is thrown horizontally with speed \(v_0\) toward the rod. The clay strikes the rod at a distance \(L\) from the pivot and sticks to it. The collision occurs almost instantaneously, and the rod-clay system swings upward after the collision.

A solid sphere of mass \(M\) and radius \(R\) is released from rest at the top of an incline of height \(H\) and rolls without slipping to the bottom. A block of mass \(M\) is released from rest at the top of an identical incline that is frictionless and slides to the bottom. If the translational speeds of the sphere and the block at the bottom of the incline are \(v_{sphere}\) and \(v_{block}\), respectively, what is the ratio \(\dfrac{v_{sphere}}{v_{block}}\)?

A small ball of mass \(m\) travels horizontally with speed \(v\) on a frictionless surface. It strikes the bottom end of a uniform rod of mass \(M\) and length \(L\) that is suspended vertically from a frictionless pivot at its top end. The ball sticks to the rod, and the ball-rod system begins to rotate about the pivot. Which of the following correctly compares the linear momentum of the ball-rod system immediately before the collision, \(p_i\), to the linear momentum of the system immediately after the collision, \(p_f\), and provides the correct physical justification?

A uniform solid cylinder of mass \(M\) and radius \(R\) is released from rest at the top of an incline of height \(h\). The cylinder rolls down the incline without slipping. Which of the following statements correctly characterizes the work done by the static frictional force and its effect on the mechanical energy of the cylinder-Earth system?

Consider the following possible justifications:
I. The force acts through a displacement.
II. The point of contact is instantaneously at rest.
III. Mechanical energy is converted to thermal energy.