Minimum Force required to move a body up along Rough Inclined Plane

When a body is on a rough inclined plane,we need to apply force to over come the component of weight acting down along the inclined plane as well as frictional force.Weight body on rough inclined plane acts in vertically downward direction.we can resolve this weight into components and SIN component acts in downward along the plane and COS component of the weight acts downward and perpendicular to the inclined plane.This provides the normal reaction on the body in the opposite direction of the weight COS component.

If we want to move the body up along the inclined plane, we shall apply a force against and in the upward direction along the inclined plane. As we try to move the body, up along the inclined plane, frictional force starts also acting the downward direction.

We need to apply a minimum force which can dominate both component of weight acting down the inclined plane and frictional force.Frictional force is the product of coefficient of friction and normal reaction.

If we apply this minimum force, body starts moving with bare minimum motion but wont't have any acceleration. But we need to apply minimum force for this to happen and hence we can show correspondingly acceleration.

If we apply a force more than this minimum force, then body starts moving up with an extra force and hence extra acceleration.

The corresponding derivation is shown here in this video below.