Cart on Frictionless Inclined Plane

Galileo used motion on an inclined plane to learn about the free fall. For us it is mainly an exercise to illustrate the properties of the normal and gravitational force and to learn how to work with Newtons laws as vector equations.

We will set ourselves the task of determining what the final velocity of the vehicle will be at the end of the incline.

Because we are interested in the motion of the vehicle we choose to apply Newtons laws to it only. The forces acting on it are

1. Force of gravity with magnitude mg
2. Normal force from surface of incline

Because of the incline these two forces are not parallel and hence gravity and normal force cannot cancel each other out. This means that there will be a net force acting on the vehicle and it will therefore necessarily undergo motion with constant acceleration. To figure the acceleration we need to obtain an expression for the net force. For this purpose it is convenient to deal with one component of force at a time. We choose the directions to be considered such that coordinate system such that

one of the directions of the coordinate system will be parallel to the net force.

Other choices are not wrong but give less elegant solutions to the problem.

Because we expect the net force to be parallel to the incline we choose the vector parallel to the incline. The two components of the net force are

where is the angle of the incline. If the brick does not sink into the plane then

This implies that

From Newton's second law this implies that the vehicle moves with constant acceleration:

Now I use my knowledge of kinematics to figure the final velocity. Remember the handy equationwhich leaves the time taken for the motion out:

Where d denotes the length of the inclined surface Lets say we start from rest at the top of the incline, then the final velocity will be

But we recognize as the height, h, of the incline so finally we get

Amusingly this is exactly the velocity after a free fall over the distance h. The length of the incline does not affect the final velocity only the time it takes to get to the end!

To test our formulae we use our beloved air-track. We have set up the photo-gate to measure the time taken for the vehicle to pass it. This time is related to the velocity by

where is the length of the vehicle. For this time we predict

We put in numbers to get

We do the experiment and get a number indistinguishable from this.