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Understanding work

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

Summary: Work is transferred energy.

Note: You are viewing an old version of this document. The latest version is available here.

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.

In order to fully appreciate the meaning of work, we consider an example. An external force is 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 and will be the subject matter of a separate module.

Figure 1: The component of external force and displacement are in same direction.
Work done on a block
 Work done on a block  (uw3.gif)

For the time being, we concentrate on work and determine its relationship with energy. Since kinetic increases by the amount of work done on the particle, it follows that work, itself, is an energy which can be added as kinetic energy to 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 illustrated in the figure below. A force is applied to retard the motion of a block. Here, the component of force is in the opposite direction to the displacement. 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, kinetic 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 :

Figure 2: The component of external force and displacement are in opposite directions.
Work done on a block
 Work done on a block  (uw4.gif)

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 are mostly restricted in its application to cases, where trajectory is linear and acceleration is constant. As a matter of fact, the set of equations (like v = u + at) relating attributes of motion is developed only for constant acceleration. In general, we limit our discussion of two dimensional motions to the cases of parabolic and circular motions only.

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. Incidentally, important forces like gravitational force and electromagnetic force are conservative in nature.

These concepts, therefore, give a huge advantage to deal with motions, which are subjected to variable force system in two or three dimensions like in the case of pendulum, spring and vertical circular motion. Also, concepts related to energy and work allow us to deal with motion even 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. This 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 have learned in the previous section, positive work means transfer of energy "to" the particle and negative work means transfer of energy "from" the particle. More simply, work is positive when component of force and displacement are in the same direction, otherwise negative.

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 formula "Frcosθ". Then, we must also recognize the directions of two vector quantities involved 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 :

Figure 3: The forces on the incline
Motion on a rough incline
 Motion on a rough incline  (uw1b.gif)

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 0.75 = 1.5 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 0.75 = 1.5 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.

Figure 4: The angle between friction and displacement
Motion on a rough incline
 Motion on a rough incline  (uw2b.gif)

W F = F r cos φ = 1.5 x 10 x cos 180 0 W = 1.5 x 10 x ( - 1 ) = - 15 J W F = F r cos φ = 1.5 x 10 x cos 180 0 W = 1.5 x 10 x ( - 1 ) = - 15 J

We must see here that computed work is the work done by friction and not by other external forces like weight or normal force. 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 :

Figure 5: Forces on the block (except external force)
Motion on a smooth incline
 Motion on a smooth incline  (uw5.gif)

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

Value of "work" is :

W g = m g sin θ x r = 10 x 10 = 100 J W g = m g sin θ x r = 10 x 10 = 100 J

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

W g = - 100 J W g = - 100 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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