Work Done: Scalar Product, Definition, Practice Problems and FAQs

Imagine yourself pushing a box on a rough surface. The force which you apply on the box tends to displace it through a small distance. A coolie carries a heavy bag on his head and heads up the stairs. Work is said to be done, since the force the coolie applies on the bag changes its position. You see two common things in both the processes - force and displacement. So, how do we define work using force and displacement? Let’s explore this here!

Table of contents

Scalar product-definition, properties
Work done by a constant force
Work done by a variable force
Difference between work and energy
Practice problems
FAQs

Scalar product-definition, properties

A vector quantity is said to have both direction and magnitude. Let us consider two vectors and . Now the scalar or dot product of these two vectors is defined as the product of their magnitude and the cosine of angle between them.

Fig. shows two vectors that are inclined at an angle between them

Mathematically, =

and represent the magnitude of the two vectors;

is the angle between.

Properties of dot product

1) Scalar product is commutative.

. = .

2) Scalar product is distributive.

For three vectors , and , we can write

.( + ) = . + .

3)( ) = (); where is a real number

For unit vectors , , along and axes, we can write

. . .

Work done by a constant force

Fig shows a constant force acting on a body

Consider a constant force acting on a body which displaces it through a displacement . The work done on the body by the force can be written as

= . = ; where is the angle between and . Here is the component of force that is acting along the displacement.

No work is done, if

(i) The angle between the force and the displacement is since .

(ii) The displacement is zero, i.e 0

(iii) If the force acting is zero.

Work is if lies between 0 and
Work is if lies between and

Work done by a variable force

A force is said to be variable when either its magnitude or direction varies. The force in terms of its individual components can be expressed as

, and represent forces in and directions.

Displacement of the body, ( , , are unit vectors along , and axes).

Work done

In the graphical sense, interprets as the area under the curve plotted with force along the vertical axis and displacement on the horizontal axis.

Note:

Work done is a scalar quantity. It carries the unit Joule and its dimensional formula is

Difference between work and energy

In physics, work and energy are interchangeable; but different quantities. Energy is the capacity to do work. Energy comes in various forms, like electrical, magnetic or thermal, but work comes into play only when a force moves an object.

Video explanation

https://www.youtube.com/watch?v=WBkQuGssftw

Practice problems

Q1) A force displaces a body through a distance of along a straight line. If the work done is , then the angle between the force and direction of the motion of the body is

Ans) (b) (c) (d)

Solution (a)

Given ; ;

Q2) A force acts on a particle that is moving in a straight line varies with distance as shown in the figure. Find the work done on the particle during its displacement from to .

Solution) Work doneArea under the force displacement graph. Hence,

Q3) Find the work done by a force, in displacing it from to

Solution) Since the force is variable, we need to integrate

Q4) The force acting on a body moving on a straight line varies with distance as , where the distance of the body varies from to . Calculate work done by the force.

(a) (b) (c) (d)

Solution)a