Manipulation of the Coefficient of Friction

Dynamics deals with the causes of motion. Galileo made many thought experiments about motion on inclined planes, and argued that if a ball coming down an incline gains velocity and a ball rolling up an incline loses velocity, then a ball rolling along a flat surface will never lose velocity, and maintain constant acceleration, if left to itself. However, we know that a ball rolling on a flat surface does slow down. This is because friction is present and opposes the relative motion of two surfaces in contact.

If friction is present on a flat surface, then it must also be present on an inclined surface. However, the main direction of motion is present due to the component of gravity acting on the object. This component is the sliding force that the ramp creates and applies on an object. Due to the fact that gravity accelerates objects coming down and decelerates objects going up, the sliding force acts in the same manner. From this we can draw that an object going up a ramp will be moving against the sliding force and frictional force present on the ramp. Likewise, when an object is going down a ramp it must be working again against friction but with the sliding force. These deductions can be better represented in equations:

Fup = Fs + Ff
Fdown = Fs - Ff

Both the sliding and frictional forces are only present in component forms because the ramp is on an incline instead of a horizontal surface where the normal and applied forces would create balance.

R is the normal force and W is the force of gravity. Using the cosine and sine rules for triangles, we can determine what the net forces acting on an object going up a ramp and coming down a ramp are. It is important to keep in mind that force is the product of mass and acceleration.

Another important equation is that the frictional force equals the coefficient (μ) times the normal force (which is simply mass times gravity). Also, the mass of an object does not change going up and down the ramp, so it...