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# Understanding work

Module by: Sunil Kumar Singh. E-mail the author

Summary: Work is transferred energy.

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Work results from the application of force on a particle, causing it to move or change its motion (velocity). In terms of mathematical notation, work is "Frcosθ". But, the important thing to know is : what is work and what is its significance? We shall attempt to answer these two questions in this module.

Work is an important concept in analyzing physical process. We must know that we have actually made transition to a different analysis framework than that of force. This different analysis framework of "work" is much simpler than force analysis and more general, in which analysis is independent of the intermediate details when work is used with other concepts like kinetic energy, work - kinetic energy theorem and conservative force system.

## Meaning of work

In physical term, application of force on a particle results in acceleration. As a result, velocity of the particle either increases or decreases. In other words, kinetic energy of the particle either increases or decreases.

We consider an external force as applied on a block such that component of force is in the direction of the displacement as shown in the figure. Here, work by force on the block is positive. The velocity and consequently kinetic energy of the block increases ( K f > K i K f > K i ). We shall know later that the kinetic energy of the block increases by the amount of work done by the force on it. This is what is known as "work - kinetic energy" theorem. This theorem will be explained in a separate module.

Since kinetic increases by the amount of work done on the particle, it follows that work, itself, is an energy which can be added to the knietic energy of the particle. The other qualification of the work is that it is the "energy" which is transferred by the force to the particle. In this sense, "work" by a force is the energy transferred "to" the particle, on which force is applied.

We, now, consider the reverse situation as shown in the figure above. The component of force is in the opposite direction to the displacement as shown in the figure above. Here, work done on the block is negative. The kinetic energy of the block decreases ( K f < K i K f < K i ) by the amount of negative work done by the external force. In this case, energy is transferred "out" of the block and is equal to the amount of negative work by force. Here, "work" by a force is the energy transferred "from" the block, on which force is applied. Thus, we can define "work" as :

Definition 1: Work
Work is the energy transferred by the force "to" or "from" the particle on which force is applied.

It is also clear that a positive work means transfer of energy "to" the particle and negative work means transfer of energy "from" the particle. Further, the term "work done" represents the process of transferring energy to the particle by the external force.

## Significance of the concept of energy and work

Application of Newton's second law of motion requires that we should have detailed information about the motion at each point of the trajectory. For this reason, we mostly restricted its application to cases, where trajectory was linear and acceleration was constant. We also limited our discussion of two dimensional motions to the cases of parabolic and circular motions only. As a matter of fact, the set of equations (like v = u + at) relating attributes of motion is developed only for constant acceleration.

On the other hand, the concept of energy is a property of state at a given instant or position and is described by the parameters of state at that point. Similarly, work involves displacement - a quantity that depends on the end points of a physical process (not on the intermediate points). Then, we have other concepts like work - kinetic energy theorem, potential energy and conservative force that we shall study in subsequent modules. These concepts allow us to make calculation based on end points without considering details of intermediate points. Mostly such simplification is possible, when force involved is conservative in nature. Incidently, important forces like gravitational force and electromagnetic force are conservative in nature.

These concepts, therefore, gives a huge advantage to deal with motions, which are subjected to variable force system like pendulum, spring force and vertical circular motion. Also, concepts related to energy and work allow us to deal with motion on irregular trajectories, provided we have the requisite work and energy information at end points. We shall illustrate the working of these concepts in appropriate contexts.

## Computation of work

Work refers to two vector quantities, but work itself is a scalar quantity. This has important implication on the way we analyze a physical process. Importantly, we can completely do away with the requirement of a coordinate system, if we wish so. It is a great simplification as far as real time application to problem situation is concerned. We shall illustrate this point of simplification about handling scalar "work" as against vectors like force, acceleration and velocity.

We must, however, also recognize that work is a signed scalar quantity. The meaning of sign attached to work is not same as in the case of vectors (force, acceleration and velocity), where sign has only meaning of direction and nothing else. The sign of work has specific meaning in relation to the physical process and is independent of coordinate directions. As we laernt in the previous section, positive work means transfer of energy "to" the particle and negative work means transfer of energy "from" the particle.

There are two equivalent ways to deal with the analysis of a process, involving work.:

### Evaluating "Frcosθ" to compute work

We first need to recognize the magnitude of force and displacement to use the formulae "Frcosθ". Then, we must also recognize the directions of two vectors to know the enclosed angle between them. Here, measurement of enclosed angle is not with respect to any reference direction like axes of a coordinate system. We come to know the sign of work by the value of "cosθ".

#### Example 1

Problem : A block of 2 kg is brought up to the top along a rough incline of length 10 m and height 5 m by applying an external force. If the coefficient of kinetic friction between surfaces is 0.1, find work done by the friction during the motion. (consider, g = 10 m / s 2 m / s 2 ).

Solution : Work is given by :

W = F r cos φ W = F r cos φ

##### Note:
We denote "φ" instead of "θ" as angle between force and displacement to distinguish this angle from the angle of incline.

Now, friction force is given as :

F = F k = μ k N = μ k m g cos θ Here, sin θ = 5 10 and cos θ = ( 1 - sin 2 θ ) F = 0.1 x 2 x 10 x ( 1 - 5 2 10 2 ) = 2 x 0.87 = 1.74 N F = F k = μ k N = μ k m g cos θ Here, sin θ = 5 10 and cos θ = ( 1 - sin 2 θ ) F = 0.1 x 2 x 10 x ( 1 - 5 2 10 2 ) = 2 x 0.87 = 1.74 N

To evaluate work in terms of "Frcosφ", we need to know the angle between force and displacement. In this case, this angle is 180° as shown in the figure.

W = F r cos φ = 1.74 x 10 x cos 180 0 W = 1.74 x 10 x ( - 1 ) = - 17.4 J W = F r cos φ = 1.74 x 10 x cos 180 0 W = 1.74 x 10 x ( - 1 ) = - 17.4 J

We must see here that computed work is the work done by friction and not by the external forces. This is reminded here just to emphasize that it is always important to elaborate the reference to the force, which is doing work.

### Using sign rule to compute work

Alternatively, we determine the component of force and multiply the same with displacement to find the value of "work". Then, we use the sign rule to assign sign to work : "if component of force is in the direction of displacement, then work is positive, otherwise negative." Let us apply this analysis framework to the problem given here,

#### Example 2

Problem : A block of 2 kg is brought up to the top along a smooth incline of length 10 m and height 5 m by applying an external force. Find work done by the gravity during the motion. (consider, g = 10 m / s 2 m / s 2 ).

Solution : In this problem, incline is smooth. Hence, there is no friction at the contact surface. Now, the component of gravity (weight) along the direction of the displacement is :

m g sin θ = 2 x 10 x 5 10 = 50 N m g sin θ = 2 x 10 x 5 10 = 50 N

Value of "work" is :

m g sin θ x r = 50 x 10 = 500 J m g sin θ x r = 50 x 10 = 500 J

We note that the component of weight is in the opposite direction to the displacement. Hence, work by gravity is negative.

W = - 500 J W = - 500 J

## Summary

We can summarize the main points of the context of work as :

1. Work is the energy transferred by the force "to" or "from" the particle on which force is applied.
2. Work by a constant force is calculated by the "Frcosθ".
3. Work by a variable force and displacement in x-direction is calculated by the integral " F ( x ) x F ( x ) x ".
4. Work by a constant force is computed by evaluating "Frcosθ". Sign of "cosθ" is the sign of work.
5. Work by a constant force is computed by multiplying scalar component of force in the direction of displacement and magnitude of displacement. Sign of the work is determined, using sign rule.
6. Work sign rule is : if component of force is in the direction of displacement, then work is positive, otherwise negative.
7. Positive work means transfer of energy "to" the particle.
8. Negative work means transfer of energy "from" the particle.

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