Work at an Angle | CAIE 9702 Paper 2 Notes

Paper 2

Physics9702

Work at an Angle

A force often points in a different direction from the motion. A rope can pull upward and forward while a crate moves horizontally.

Only the component of the force parallel to the displacementtransfers energy through that displacement.

Keep only the component along the motion

A hand-drawn blue crate is pulled along a horizontal floor by a rope that slopes upward. — The rope pulls upward and forward, but the crate moves horizontally. The exact component diagram below shows why only the forward component transfers energy through this displacement.

The box moves horizontally, so only the horizontal part of the pull, F cos 30°, does work: W = F s cos 30°. The vertical part does no work on the box.

W = Fs\cos\theta

Symbols

$W$ = work done by the force (J)
$F$ = magnitude of the force (N)
$s$ = displacement (m)
$\theta$ = angle between force and displacement (° or rad)

At 0°, force and motion are together: the work is positive and has its greatest value.
At 90°, the force is perpendicular to the motion: the work done by that force is zero.
At 180°, the force opposes the motion: its work is negative, so it removes energy from the object.

Worked example

An angled pull

A 200 N rope pulls a crate 5.0 m horizontally. The rope is 30° above the horizontal. Find the work done by the tension.

The angle between the tension and displacement is 30°.
$W=200\times5.0\times\cos30^\circ=866\text{ J}$ .

Answer

8.7\times10^2\text{ J}

Common mistake

Do not automatically use the full path length. For the work done by weight, use the vertical displacement because weight acts vertically.

Your turnExam-style

A 12 N friction force acts on a sliding box for 3.0 m. Calculate the work done by friction and state what its sign means.

Show worked answer

Worked answer: $W=12\times3.0\times\cos180^\circ=-36\text{ J}$ . The negative sign means friction transfers 36 J away from the box's mechanical energy store.