Question

show that the work done by a force is given by the product of the force and the component of displacement along the force

Answer 1

force is said to do work when it acts on a body so that there is a displacement of the point of application in the direction of the force. Thus, a force does work when it results in movement.

The work done by a constant force of magnitude F on a point that moves a displacement ΔxΔx in the direction of the force is simply the product

W=F⋅ΔxW=F⋅Δx

In the case of a variable force, integration is necessary to calculate the work done. For example, let’s consider work done by a spring. According to the Hooke’s law the restoring force (or spring force) of a perfectly elastic spring is proportional to its extension (or compression), but opposite to the direction of extension (or compression). So the spring force acting upon an object attached to a horizontal spring is given by:

Fs=−kxFs=−kx

that is proportional to its displacement (extension or compression) in the x direction from the spring’s equilibrium position, but its direction is opposite to the x direction. For a variable force, one must add all the infinitesimally small contributions to the work done during infinitesimally small time intervals dt (or equivalently, in infinitely small length intervals dx=vxdt). In other words, an integral must be evaluated:

Answer 2

Answer:

Let F be the force and s the displacement. Let θ be the angle between F and s. Geometrically, s cos θ is the projection of vector s along vector F. Hence Work done by a force is nothing but the product of the magnitude of foce and the projection of displacement along the force.