Question

Explain with graphs the difference between work done by a constant force and by a variable force.

Answer 1

Work done by a constant force: When a constant force F acts on a body, the small work done (dW) by the force in producing a small displacement dr is given by the relation,

dW = (F cos θ) dr ..(1)

The total work done in producing a displacement from initial position r_i to final position r_f is,

W = `int_{r_i}^{r_f}`dW ..............(2)

W = `int_{r_i}^{r_f}`(F cos θ)dr = (F cos θ) `int_{r_i}^{r_f}` dr = (F cos θ) (r_f - r_i) ...............(3)

The graphical representation of the work done by a constant force is shown in the figure given below. The area under the graph shows the work done by the constant force.

 

Work done by the constant force

Work done by a variable force: When the component of a variable force F acts on a body, the small work done (dW) by the force in producing a small displacement dr is given by the relation
dW = F cos θ dr [F cos θ is the component of the variable force F]
where F and θ are variables. The total work done for a displacement from initial position r_i to final position r_f is given by the relation,

W = `int_{r_i}^{r_f}`dW = `int_{r_i}^{r_f}`F cos θ dr ..................(4)

A graphical representation of the work done by a variable force is shown in the figure given below. The area under the graph is the work done by the variable force.

 

Work done by a variable force