13.1 Work and energy  (Page 4/5)

Example

Problem 1: A block of mass "10 kg" slides down through a length of 10 m over an incline of 30°. If the coefficient of kinetic friction is 0.5, then find the work done by the net force on the block.

Solution : There are two forces working on the block along the direction of displacement (i) component of gravity and (ii) kinetic friction in the direction opposite displacement.

The component of gravity along displacement is "mg sinθ" and is in the direction of displacement. Work by gravity is :

$$\begin{array}{l}{W}_G=mgL\sin 30°=10X10X10X\frac{1}{2}=500J\end{array}$$

Friction force is " ${\mu }_{K}mg\sin \theta$ " amd is in the opposite direction to that of displacement.

$$\begin{array}{l}{W}_F=-{\mu }_{K}mgL\cos 30°=-0.5X10X10X10X\frac{\mathrm{\surd }3}{2}=-433J\end{array}$$

Hence, work by net force is :

$$\begin{array}{l}{W}_{\mathrm{net}}=W_G+W_F=1000-433=567J\end{array}$$

The important thing to note here is that reference of component of force is with respect to displacement - not with respect to any coordinate direction. This is how we eliminate the requirement of coordinate system to calculate work.

Imgine if we first find the net force. It would be tedius as friction acts along the incline, but gravity acts vertically at angle with the incline. Even if, we find the net force, its angle with displacement would be required to evaluate the expression of work. Clearly, calculation of work for individual force is easier. However, we must keep in mind that we can employ this technique only if we know the forces beforehand.

Anaysis of motion

The concepts of work and energy together are used to analyze motion. The basic idea is to analyze motion such that it does not require intermediate details of the motion like velocity, acceleration and path of motion. The independence from the intermediate details is the central idea that makes work - energy analysis so attractive and elegant. It allows us to analyze motion of circular motion in a vertical plane, motion along paths which are not straight line and host of other motions, which can not be analyzed with laws of motion easily. This is possible because we find that work by certain class of force is independent of path. Further, under certain situations, the analysis is independent of details of attributes like velocity and acceleration.

Path of motion

In this section, let us examine the issue of path independence. Does the work depend on the path of motion? The answer is both yes and no. Even though computation of work involves displacement - a measurement in terms of end points, work is not always free of the path involved. The freedom to path depends on the nature of force. We shall see that work is independent of path for force like gravity. Work only depends on the vertical displacement and is independent of horizontal displacement. It is so because horizontal component of Earth's gravity is zero. However, work by force like friction depends how long (distance) a particle moves on the actual path.

The class of force for which work is independent of path is called "conservative" force; others are called "non-conservative". We can, therefore, say that work is independent of path for conservative force and is dependent for non-conservative force. This is the subject matter of a separate module on "conservative force" and as such we would not elaborate the concept any more.