A Cart and a Block on a Ramp

Newton’s second law states that the net force on an object equals its mass times its acceleration, F = ma. On an inclined plane that makes an angle θ with the horizontal, it is convenient to separate the weight mg into two components: one parallel to the surface, mg sin θ, which pulls the object down the ramp, and one perpendicular to the surface, mg cos θ, which is balanced by the normal force N. When an object slides, kinetic friction acts up the ramp with a magnitude f = μN, where μ is the coefficient of kinetic friction for the two surfaces.

Energy provides a second way to analyze motion. The work done by a force F acting over a distance d along the direction of motion is W = Fd. The work done by all forces on an object equals its change in kinetic energy, and the work done against friction is converted to thermal energy. For this passage, use g = 10 m/s², sin 37° = 0.6 and cos 37° = 0.8.

Part 1. A student releases a 2 kg cart from rest at the top of a smooth ramp inclined at 37°. The wheels are well lubricated, so friction can be neglected. A motion sensor records the distance the cart has traveled along the ramp at several times. Table 1 shows the results.

Table 1. Distance traveled by the cart on the smooth ramp.

Part 2. The student replaces the cart with a 2 kg wooden block and covers the ramp with three different surfaces, X, Y and Z. For each surface, the block is placed on the ramp and the acceleration is measured as the block moves down. For surface Z, the block is first given a small push because it does not start to slide on its own; once it is moving, its speed stays constant. Table 2 shows the results.

Table 2. Motion of a 2 kg block on three surfaces on the 37° ramp.

The student repeats the measurements several times for each surface and finds that the results are reproducible to within a few percent. She also checks that the ramp angle does not change when the block is placed on it, and that the sensor’s reading of the start time matches the release of the block.

On a smooth surface, the only forces on the cart are its weight and the normal force from the ramp, and the normal force does no work because it is perpendicular to the motion. The student therefore expects the acceleration of the cart not to depend on its mass, and she wants to check this by repeating Part 1 with different masses.