Show that a force that does no work must be a velocity dependent force.

Explanation

Let `vec F` be the force acting on an object moving with velocity `vec v `, causing a displacement ` vec (dl) ` for time dt.
Then work done is given by :
` dW = vec F * vec (dl) `
` = vec F * vec v dt`
Given , `dW =0 `
` rArr " " vec F * vec v dt = 0 `
or `F v cos theta = 0 `
or ` theta = 90^(@)`
Therefore, for no work is done on the object, `vec F` must be velocity-dependent if angle between `vec F " and "vec v " is " 90^(@)` . If the direction of velocity `vec v` changes, `vec F` must also change so that angle between them remains `90^(@)`.
It happens when a charged particle is moving in magnetic field.