More advanced particle on slopes questions involve friction or extra external forces.

Example:

A sledge of mass 78 kg is pulled up a slope by means of a rope. The slope is modelled as a rough plane inclined at an angle α to the horizontal, whereThe rope is modelled as light and inextensible and is in a line of greatest slope of the plane. The coefficient of friction between the sledge and the slope is 0.25. Given that the sledge is accelerating up the slope with acceleration

(a) find the tension in the rope.

The rope suddenly breaks. Subsequently the sledge comes to instantaneous rest

and then starts sliding down the slope.

(b) Find the acceleration of the sledge down the slope after it has come to

instantaneous rest.

First draw a diagram.

Ifwe can findandby use of a right angled triangle:

Resolve perpendicular to plane:

Resolve parallel to plane:

b)

When the sled comes to rest and starts accelerating down the slope the direction of friction changes to act up the slope. We resolve perpendicular and parallel to the slope.

Resolve perpendicular to plane:

Resolve down the slope: