A solid disk pulley of mass \(M\) and radius \(R\) is mounted on a fixed, frictionless, horizontal axle. The rotational inertia of the pulley is \(I = \frac{1}{2}MR^2\). A light string of negligible mass is wrapped around the right side of the pulley and attached to a hanging block of mass \(m\). A stationary brake pad is pressed against the left side of the pulley’s outer rim. The brake pad exerts a constant kinetic frictional force of magnitude \(f_b\) on the pulley. The system is released from rest, and the block falls downward, causing the pulley to rotate.

A technician uses a wrench to loosen a stuck bolt. The wrench handle has a length \(L\) and is oriented along the positive \(x\)-axis, with the center of the bolt at the origin. The technician applies a force of constant magnitude \(F\) to the end of the handle at \(x = L\). Which of the following diagrams represents the force orientation that produces the maximum torque magnitude about the center of the bolt?

A mechanic uses a wrench of length \(r = 0.20 \text{ m}\) to tighten a bolt on a machine. The mechanic applies a force of varying magnitude at the very end of the wrench handle. The magnitude of the applied force \(F\) as a function of the angle \(\theta\) between the wrench handle and the force vector is shown in the graph.

What is the magnitude of the torque exerted on the bolt when the angle \(\theta\) is \(30^\circ\)?

A technician uses a wrench to loosen a bolt. In the first attempt, the technician applies a force of magnitude \(F\) at a distance \(r\) from the center of the bolt, such that the force makes an angle of \(30^{\circ}\) with the wrench handle. In the second attempt, the technician applies a force of the same magnitude \(F\) at a distance \(2r\) from the center of the bolt such that the force is perpendicular to the wrench handle. What is the ratio of the magnitude of the torque produced in the second attempt to the magnitude of the torque produced in the first attempt?

In Scenario 1, a block of mass \( M \) is suspended at rest from an ideal string attached to a fixed horizontal beam. In Scenario 2, the same block is suspended from an identical string that is wrapped around a uniform disk-shaped pulley of mass \( M \) and radius \( R \). The pulley is mounted on a horizontal, frictionless axle. The block in Scenario 2 is released from rest and allowed to accelerate downward as the pulley rotates. Let \( T_1 \) be the tension in the string in Scenario 1 and \( T_2 \) be the tension in the string in Scenario 2. Which of the following correctly compares the tensions and provides a valid justification?

A student is provided with four different objects, each having the same total mass \(M\) and the same outer radius \(R\). The objects are a thin hoop, a solid disk, a solid sphere, and a thin hollow spherical shell. Each object is rotated about an axis passing through its center of mass. Which of the following correctly ranks the rotational inertia \(I\) of the four objects?

A block of mass \( m \) is attached to the bottom of a light vertical spring of spring constant \( k \) that is suspended from a ceiling. The block is gently lowered until it hangs at rest at its equilibrium position. The spring stretches a vertical distance \( y_0 \) from its unstretched length to reach this equilibrium position. The downward direction is defined as positive, and \( y = 0 \) is defined as the position of the block when the spring is at its unstretched length.

A small sphere of mass \(m\) is attached to a string of length \(L\) and displaced to an angle \(\theta = 30^\circ\) from the vertical, as shown in the figure. The sphere is then released from rest. Which of the following diagrams correctly represents the actual forces (solid arrows) and the components of the gravitational force resolved along and perpendicular to the string (dashed arrows) at the instant of release?

A block of mass \(m\) is attached to the lower end of a vertical ideal spring with spring constant \(k\). The upper end of the spring is fixed to a ceiling. The block is initially at rest at its equilibrium position. A student pulls the block down a distance \(A\) and releases it from rest, causing the block to oscillate in simple harmonic motion. Which of the following expressions represents the maximum magnitude of the acceleration of the block?

An object is attached to an ideal horizontal spring and is undergoing simple harmonic motion on a frictionless surface. A position-time graph for the object is shown below, where the object is released from rest at the maximum positive displacement \(x = +A\) at time \(t = 0\). Which of the following graphs best represents the velocity \(v\) of the object as a function of time \(t\)?