Physics on an Incline Plane

A block of mass m rests on a surface that is inclined at an angle q relative to the horizon.

Frame of Reference:

• The key to solving problems involving an incline plane is to resolve the force-vectors acting on the block (or blocks) into components that are either parallel or perpendicular to the surface of the incline plane. Since the block is assumed not leave the incline plane's surface all the motion will take place along the incline plane - the block's acceleration perpendicular to the plane is always zero as long as it is in contact with the incline plane.
• This is equivalent to rotating a normal coordinate system by an angle q.

Component of gravity pushing the block down the incline plane.
Wx'
= mg sin(q)

Component of gravity pushing the block against the incline plane.
Wy'
= mg cos(q)

Relevant Physics:

• As long as a block stays in contact with the surface of the incline plane, the sum of the forces on that block that are perpendicular to the incline plane will be equal to zero and the motion of the block will be along the surface of the incline plane. Newton's Second Law then gives the general relation that is always true for this type of problem.

Frictionless Incline Plane with no Applied Force:

• In this case the block would accelerate down the incline plane. In fact if the incline plane was vertical so that q = 90o then a = -g sin(90o) = -g, i.e. then problem would go over to that of a simple free-fall problem.

Frictionless Incline Plane with Parallel Applied Force:

Free Body Force Diagram

• Here the block could accelerate up or down the incline plane depending upon the size of the applied force.

Incline Plane with Parallel Applied Force and Friction:

Free Body Force Diagram

• If the applied force is not parallel to the surface then it will need to be split into two components, one parallel to the surface and the other perpendicular to the surface. Note that this will cause the frictional force to change since the normal force will be increased or decreased.
• The direction of motion depends not only upon the applied force, the weight of the block, and also on the direction of the force of friction.

Block on an Incline Plane
A block is pushed up an incline plane by an force parallel to the incline plane. The coefficient of friction between the block and the incline plane, the applied force, the angle of the incline plane, and the mass of the block can be varied. Displayed are the resulting forces acting on the block as it moves up or down the incline plane. QT Movie