Work is said to be done only when an applied force succeeds in moving a body.
Consider a wooden block which is at rest on a horizontal surface.
When the block is pulled by an external horizontal force “F
it is put into motion and is displaced through a certain distance “d
” and we say work is done on the block.Let’s consider another case, where the direction of the force is not along the direction of the motion of the block. The force makes an acute angle q with the horizontal.
In this case too, we observe that the block moves in the horizontal direction only.
Here, the total amount of force is not being utilised to move the block. But
Only the component of the applied force along the direction of the displacement is responsible for the motion of the block.
This component is F cosq.
Thus here, the work done on the block by the force is the product of F cos q and displacement.
In this expression for the work done, both F and d are vectors whereas the product, work done, is a scalar.
Thus, this quantity ‘F d cosq’ is called the scalar product of the two vectors F and d.
In vector form, it is denoted by a dot in between the two vectors, and is read as F dot d. Hence it is also referred to as the dot product.
The dot product of two vectors A and B is equal to the product of the magnitude of the two vectors and cosine of the angle between the two vectors. The result of a dot product is always a scalar.
Now let’s see some properties of the dot product of vectors.
1. The dot product of vectors is always a scalar quantity.
2. The dot product is commutative.
As the dot product is commutative,
Now, using the dot product we can measure the work done as the dot product of the force and displacement, which is equal to F d cosq.
However, the work done can be positive, negative or zero
depending on the values of the angle between the force and the displacement, q.
The work done by a force is positive
when the angle between force and displacement is acute, i.e., the angle is less than 90 degrees.
An example of this is when an apple falls freely towards the earth. The work done by the gravitational force on the apple is positive. In this case, the angle between the displacement of the apple and the gravitational force is zero.
Similarly, when a spring is compressed, the work done by the compressing force is positive.
In this case too, the angle between the direction of the applied force and displacement, which is the compression of the spring, is zero.
The work done by a force is negative
when the angle between force and displacement is obtuse, that is, greater than 90 degrees and less than or equal to180 degrees.
In the case of a rising balloon as shown, the work done by the force of gravity acting on the balloon is negative since it is opposite to the displacement of the balloon.
In the example of a boy lifting a stone, the work done by the gravitational force on the stone is negative as the displacement is against the gravitational pull
The work done by a force is zero
when the force is perpendicular to the displacement.
Consider a block moved over a horizontal surface. In this case, the displacement is along the horizontal whereas the gravitational force, which is the weight of the block, acts vertically downwards. Here work done by the gravitational force is zero since it acts perpendicular to the displacement of the body.
As another example, consider a stone tied to a string whirled in a circular path.
Here, a centripetal force acts on the stone to maintain its circular path.
Centripetal force is always directed towards the centre of the circular path whereas the displacement of the stone for an infinitesimal time interval is along the tangent to the circle at the point under consideration.
As the direction of the force is perpendicular to the displacement, the work done by a centripetal force is zero.
The SI unit of work is joule which is denoted by the letter
J, named after the famous British scientist James Prescott Joule.
one joule is defined as the work done on a body when a force of one newton displaces the body in the direction of the force by one metre.
Now you will learn about the relationship between work done on a body by a net force and the change in its kinetic energy by using the Work-Energy Theorem.
According to the Work-Energy Theorem, the work done by a net force on an object is equal to the change in its kinetic energy can be proved mathematically as the following..