Path Independence

Part A

Consider three ways to get a block from height h_i down to height h_f = 0: dropping the block straight down through the air (freefall), sliding the block down a frictionless ramp, or sliding the block along a frictionless track. The options are illustrated below.

Assuming that the block starts from rest, what is the final speed of the block as it reaches the ground in each case?

Solution: We begin by solving all three problems using mechanical energy. We will then re-solve the first two cases using the equations of kinematics and dynamics to illustrate the agreement between the methods.

Method 1

System: Block as point particle plus the earth as a rigid body with infinite mass.

Interactions: In each case, there is a conservative gravitational interaction between the block and the earth which will provide a gravitational potential energy. In the final two cases (the ramp and the track) the block is also subject to a [non-conservative] normal force from the surface upon which it travels. In each case, however, the normal force does no work, since it is always perpendicular to the direction of the block's travel (the block is always moving parallel to the surface and the normal force is, by definition, perpendicular to the surface). Thus, in all three cases, the non-conservative work is zero.

Model: [Mechanical Energy and Non-Conservative Work].

Approach: Since the non-conservative work is zero, the mechanical energy will be constant:

In each case, the initial-state final-state diagram and [energy bar graphs] will be:

In each case, the constant energy relationship is:

giving:

independent of the path taken from h_i to the zero height level.

Method 2

System: Box as point particle.

Interactions: External influences from the earth (gravity) and for the block on the ramp, the ramp provides a normal force.

Model: One-Dimensional Motion with Constant Acceleration and Point Particle Dynamics.

For the drop and the ramp, we can recover the same result using kinematics.

For the drop, if we set up coordinates with the positive y direction pointing vertically, the acceleration will be a_y = – g, so that the most straightforward approach is to use the Law of Change:

Using the givens:

Substituting these givens yields:

For the ramp, we tilt the coordinates to place the x-axis along the ramp surface. The free body diagram for the box is:

which implies that Newton's 2nd Law for the x direction takes the form:

giving:

With the acceleration determined, we turn to kinematics to find the final velocity. We have the givens:

We then substitute into the Law of Change for one-dimensional motion:

Substituting the givens yields:

Part B

In Part A, we showed that the speed of the blocks at the end of their trip is independent of the path taken from one height to another. This equality does not mean, however, that the trips are identical in every way. The time taken to complete the trip will depend upon the details of the path. We can show this by calculating the time taken to complete the trip for the case of the dropped box and the box sliding down the ramp using kinematics.

System: The box as point particle.

Interactions: External influences from the earth (gravity) and for the block on the ramp, the ramp provides a normal force.

Model: One-Dimensional Kinematics with Constant Acceleration.

Approach: One direct way (of several) to obtain the time is to use the Law of Change:

which gives: