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Conservative force is the work done by the force that depends on the initial and the final position of the object and independent of the path a body has covered.Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Image will be uploaded soon]

So, the work done at point A, i.e. WA = Work done at B, i.e. WB

A nonconservative force is a work done by the force that considers the path traced with the initial & final position of the object.

In this article, we will study what are conservative and nonconservative forces through illustrative examples.

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A body displaces from A to B by â€˜lâ€™, and the force acting at point A is mg. So, the work done will be:

WAB = mg . l. Cos (90 + Ï´) = - mglSinÏ´ â€¦..(1)

Similarly, from A to C, the work done will be:

WAC = mglCos90Â° = 0â€¦.(2), and

From C to B, the length component along CB is â€˜lSinÏ´â€™ and the work done is:

WBC = - mg(lSinÏ´) Cos180Â° = - mglSinÏ´ â€¦(3)

Here, Ï´ = 180Â° because the motion of the body is against gravity.

We can see that WAB = WBC (Independent of the path). The work done by the force is conservative.

Letâ€™s take another example:

Consider a block of mass 10 kg falling from 10 m height. The work done by the conservative force will be:

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W = F . d CosÏ´

F = mgÂ

â‡’ 10 x 9.8 = 98 N

Here, the angle between the gravity and the displacement is 0Â° because both of these are acting downwards. So, the work done will be:

W = 98Â x 10Â x Cos0Â° = 980 J

Now, letâ€™s say this block moves up and then down.

Case1: The work done, W1 = 980 N

Case 2: When moves up

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Image will be uploaded soon]

Work done against the gravity, W2 = 98Â x 10Â x Cos180Â°= - 980 JÂ

Case 3: Ball drops

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Image will be uploaded soon]

Here, the work done, W3Â = 980 N

âˆ´ The total work doneÂ = 980 - 980 + 980 = 980 N

Here, we can see that the gravitational force is the conservative force.Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Image will be uploaded soon]

The work done by the spring force depends on the initial and the final position and not on the path traced by the spring. Thatâ€™s why spring forces are called the conservative forces.Â

Similarly, the work done by a conservative force in a closed path is zero.

Nonconservative force is a type of force whose work done relies on the path, not on the initial and the final position.

For example,

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If we go along a 5 m path, the frictional force will be less as compared to 7 m long path. This means that work done at initial & final points, i.e., WI â‰ WII.Â

Similarly, work done by a conservative force in the closed path is not zero.

Letâ€™s see some questions to identify a conservative and a nonconservative force:

Question 1: If the force, F = 5yi + 5xj acts on the body, then what is true for F?

Itâ€™s a conservative force

It is a nonconservative force

Depends upon x

Depends upon both x and y

Objective Answer: (a): F is a conservative force.

Explanation:

We will write this equation as: F = Fxi +Â Fy j + 0 (Fzk = 0)....(p)

For a force to be conservative, the following three conditions are necessary:

Condition 1: \[\frac{âˆ‚Fx}{âˆ‚y}\] = \[\frac{âˆ‚Fy}{âˆ‚x}\]Â

Condition 2: \[\frac{âˆ‚Fx}{âˆ‚z}\] = \[\frac{âˆ‚Fz}{âˆ‚x}\]Â

Condition 3: \[\frac{âˆ‚Fy}{âˆ‚z}\] = \[\frac{âˆ‚Fz}{âˆ‚y}\]Â

Now, differentiating equation (p) w.r.t. x, y, and z individually, to check if these conditions are true or not.

\[\frac{âˆ‚Fx}{âˆ‚y}\] = \[\frac{âˆ‚Fy}{âˆ‚x}\] = 5

\[\frac{âˆ‚Fx}{âˆ‚z}\] = \[\frac{âˆ‚Fz}{âˆ‚x}\] = 0

\[\frac{âˆ‚Fy}{âˆ‚z}\] = \[\frac{âˆ‚Fz}{âˆ‚y}\] = 0

By matching with three conditions mentioned above, we can see that the force is conservative.

Question 2: If a force acts on the body, and it follows a path as 2yi + 3xj; which among the following statements is true?Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Conservative force

Nonconservative force

Depends upon x

Depends upon both x and y

Objective Answer: (b): Nonconservative force

Explanation:Â

We can determine the condition for the force F by using the Condition 1: \[\frac{âˆ‚Fx}{âˆ‚y}\] = \[\frac{âˆ‚Fy}{âˆ‚x}\]

\[\frac{âˆ‚Fx}{âˆ‚y}\] = 2, \[\frac{âˆ‚Fy}{âˆ‚x}\] = 3,Â

Here, 2 â‰ 3, which means the force is nonconservative.

Question 3: If a force F = 3x2i + 45j acts on the body and follows a path as shown below. Calculate the WNET (Oâ†’ Aâ†’ Bâ†’ Câ†’ O).

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Image will be uploaded soon]

Solution: F = 3x2i + 5y2j=Â Fxi + Fy j + 0

Where,Â

Â Fx Â = 3

Â Fy = 5

Applying the conditions again, we get,

\[\frac{âˆ‚Fx}{âˆ‚y}\] = \[\frac{âˆ‚Fy}{âˆ‚x}\] = 0 = 0Â

Here, the differentiation of the constant value of â€˜xâ€™ w.r.t. â€˜yâ€™ is zero and vice-versa. The same is for other conditions.

\[\frac{âˆ‚Fx}{âˆ‚z}\] = \[\frac{âˆ‚Fz}{âˆ‚x}\] = 0 = 0

\[\frac{âˆ‚Fy}{âˆ‚z}\] = \[\frac{âˆ‚Fz}{âˆ‚y}\] = 0 = 0

We can see that in a closed-loop, the net work done is zero.

FAQ (Frequently Asked Questions)

Question 1: The work done during the zero displacements can never be zero. Justify this statement.

Answer: Consider a block displaced by â€˜dâ€™ meters from point A to B. A friction force acting on it is f, then work done by the frictional force will be :

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â W = - f. dâ€¦.(1)

If the same block displaced from B to A, then again W = - f. d â€¦.(2)

So, the net work done = - f. d + (- f. d) = - 2f. d.

Hence,Â W_{NET} â‰ 0 during zero displacements.

Question 2: Why is the potential energy defined for the Conservative Force?

Answer: Potential energy of a system is due to the shape, position, and configuration stored in the system. It relies on the initial and the final point only; that's why it is defined for the conservative force.

Question 3: Can the work done by a Conservative Force be negative?

Answer: Work done by a conservative force can either be positive or negative.Â

If the work done by the force is the direction of the displacement of the object, then the work done is positive.Â

If the work done by the force is in the opposite direction to that of the displacement of the object, then the work done by the force is negative.

Question 4: Is tension a Nonconservative Force?

Answer: Yes, tension is a nonconservative force.