A student investigates the energy transformations that occur when a solid bouncy ball of mass \(m = 0.050 \text{ kg}\) collides with a rigid force sensor. The sensor is connected to a computer that records the force exerted on the sensor over time. The solid plate of the force sensor behaves mathematically like an ideal spring with a very large, unknown spring constant \(k\).

The student drops the ball from rest from an initial height \(h\) directly above the sensor. During the collision, the sensor’s surface compresses by a maximum distance \(d\) and records a maximum impact force \(F_{max}\). Because the sensor is extremely stiff, the compression distance \(d\) is microscopic, meaning the change in the ball’s gravitational potential energy during the compression itself is negligible.

A block of mass \(M_1\) is connected to a block of unknown mass \(M_2\) by a light string with an ideal spring scale of negligible mass spliced into the middle. The system is pulled vertically upward by an external force of magnitude \(F\), causing the blocks to accelerate upward. The values of \(M_1\), the external force \(F\), and the spring scale reading \(F_s\) are known. Which of the following additional measurements, if any, must be made to determine the unknown mass \(M_2\)?

A small cart is released from rest at time \( t = 0 \) and moves down a straight track with constant acceleration \( a \). A motion sensor measures the speed \( v \) of the cart at different positions \( x \) along the track, where \( x = 0 \) is the starting position. The data collected is used to create the graph shown of the square of the speed \( v^2 \) as a function of position \( x \). Based on the graph, what is the acceleration of the cart?

A uniform object of mass \(M\) and radius \(R\) is released from rest at the top of a ramp inclined at an angle \(\theta\) to the horizontal. The object rolls without slipping as it moves down the ramp. The rotational inertia of the object is given by the expression \(I = cMR^2\), where \(c\) is a dimensionless constant. Which of the following expressions correctly represents the distance \(d\) the object travels along the ramp as a function of time \(t\)?

A student is calibrating a projectile launcher to be used in a laboratory experiment. The launcher is placed on the ground and set to fire a ball at an angle \( heta \) above the horizontal across a level field. In the first calibration trial, the ball is launched with an initial speed \( v_0 \) and travels a horizontal distance \( D_1 \) before hitting the ground. In the second calibration trial, the launcher is adjusted to fire the ball at the same angle \( heta \) but with an initial speed of \( 2v_0 \), resulting in a horizontal distance \( D_2 \). What is the ratio of the horizontal distance \( D_2 \) to the horizontal distance \( D_1 \)?

A block of known mass \(M\) rests on a rough horizontal table. It is attached to a lightweight, inextensible string that passes over an ideal pulley at the edge of the table. A mass hanger of negligible mass is attached to the other end of the string. The student is provided with a set of varying known masses \(m\) that can be added to the mass hanger, a ruler, and a stopwatch. The student is tasked with designing experiments to determine both the coefficient of static friction \(\mu_s\) and the coefficient of kinetic friction \(\mu_k\) between the block and the table.

A student is testing the tensile strength of a rigid rod of negligible mass. The student attaches Force Sensor 1 to the left end of the rod and Force Sensor 2 to the right end. The student pulls Force Sensor 1 to the left with a force of magnitude \( F \) while an assistant pulls Force Sensor 2 to the right with a force of magnitude \( F \), such that the rod remains at rest. Which of the following free-body diagrams best represents the horizontal forces exerted on the rod?

A student conducts an experiment to determine the constant acceleration \(a\) of a cart that starts from rest and rolls down a long, straight track. The student uses a sensor to collect data for the distance \(d\) the cart travels as a function of time \(t\) since it was released. To determine the acceleration using the slope of a linear graph, the student should plot which of the following quantities on the vertical and horizontal axes?

A student conducts an experiment to determine the local acceleration due to gravity, g, by measuring the period T of a simple pendulum for several different string lengths L. The student intends to linearize the data so that the value of g can be calculated from the slope of a best-fit line. Which of the following combinations of quantities, when plotted on the vertical and horizontal axes, would result in a linear graph with a slope S such that g = “4\pi^2 / S”?

A uniform solid cylinder of mass \(M\) and radius \(R\) has a light string wrapped multiple times around its circumference. The free end of the string is attached to a fixed horizontal ceiling. The cylinder is held at rest and then released, falling vertically as the string unwinds without slipping. The rotational inertia of a solid cylinder about its center of mass is \(I = \dfrac{1}{2}MR^2\). After the center of mass of the cylinder has fallen a vertical distance \(h\), what is the speed of the center of mass of the cylinder?