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.

A uniform circular disk with a radius of \(0.060 \text{ m}\) is rotating about its center at a constant angular velocity of \(20 \text{ rad/s}\). What is the tangential speed of a point located on the outer edge of the disk?

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?