4.1 Work Done by a Constant Force

4.1 Work Done by a Constant Force

Let us consider an object which is being pulled by a constant force $\mathbf{F}$. The force displaces the object through a displacement $\mathbf{d}$ in the direction of force. In such a case, work $W$ is defined as the product of the magnitude of the force $F$ and magnitude of the displacement $d$. This can be written as:

Equation (4.1) shows that if the displacement is zero, no work is done even if a large force is applied. For example, pushing on a wall may tire your muscles, but work done is zero as shown in Fig. 4.1.

The force applied on a body may not always be in the direction of force as shown in Fig. (4.2). If the force $\mathbf{F}$ makes an angle $\theta$ with the displacement $\mathbf{d}$ (Fig. 4.3), the work done is equal to the product of the component of force along the direction of the displacement and the magnitude of displacement. Then

or

Equation (4.3) shows that work is a scalar quantity.

The unit of work is joule (J). From Eq. (4.1), we have

When a constant force acts through a distance $d$, the event can be plotted on a simple graph (Fig. 4.4). The distance is normally plotted along $x$-axis and the force along $y$-axis. As the force does not vary, in this case, the graph will be a horizontal straight line. If the constant force $F$ (newton) and the displacement $d$ (metre) are in the same direction, then the work done is $Fd$ (joule). Clearly shaded area in Fig. 4.4 is also $Fd$. Hence, the area under a force-displacement curve can be taken as to represent the work done by the constant force. In case the force $F$ is not in the direction of displacement, the graph is plotted between $F\cos\theta$ and $d$.

From the definition of work, we find that:

(i) Work is a scalar quantity.

(ii) If $\theta < 90^{\circ}$, work is done and it is said to be positive work.

(iii) If $\theta = 90^{\circ}$, no work is done.

(iv) If $\theta > 90^{\circ}$, the work done is said to be negative.

(v) SI unit of work is $\mathrm{N}\cdot\mathrm{m}$, also known as joule (J).