Comprehensive Question 1: Define work and show that it is the dot product of force and displacement. At what conditions, work done will be maximum or minimum?
In Physics, work is done only when an object is moved through some displacement while a force is applied to it.
Definition of work
When we apply some force on a body and the body moves through some displacement in the direction of force, then the product of force and displacement is called work. Mathematically,
Unit of work
Unit of work can be derived from the above equation.
Unit of work = unit of force × unit of displacement = N m = J.
It can be defined as, work is said to be 1 J if a force of 1 N displaces a body by 1 m in the direction of force.
Dimensions of work
Dimensions of work = dimensions of force × dimensions of length = |M1L1T-2||L1| = |M1L2T-2|.
Work as dot product of force and displacement
Let a person is pulling a box with a force . As the box is dragged on the ground, it covers a displacement . Further suppose the angle between force and displacement is θ. Resolve into rectangular components and . Now the y-component of the force is perpendicular to the displacement which means it has no contribution in pulling the box. Similarly, the x-component is along the displacement of the box. Therefore, the work is done entirely by this component of the force. Thus, by the definition of work as the product of displacement and the force along the displacement,
Fdcosθ = .=dot product of force and displacement
Therefore, we can say that work is the scalar quantity equal to the dot product of force and displacement.
Maximum and Minimum Work
Looking into the equation of work, W = Fdcosθ, we see work is directly proportional to the force and displacement. Increasing force or displacement or both of them will increase the quantity of work and vice-versa. Similarly, work also depends on the cosine value of the angle between force and displacement vectors. For constant force and displacement, therefore, work will depend on the angle between force and displacement. If θ = 0°, cosθ = 1. This is the maximum value of the cosine of an angle. Therefore, work done is maximum. Similarly, if θ = 90°, cosθ = 0 and work will be zero. For θ = 180°, cosθ = – 1 and the numerical value of the work done is negative or minimum. Hence, for a constant force, work done will be maximum if force and displacement are in the same direction and minimum if the force and displacement are in the opposite directions.