Work, Energy and Power

Physics Part I

Physics

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Is work a scalar or a vector quantity?

Work is dot product of two vectors. i.e. .

And dot product is a scalar quantity. Therefore, work is a scalar quantity.

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A uniform chain of length L and mass m is lying on a smooth table and one-nth part of its length is hanging vertically down over the edge of the table. Find the work required to pull the hanging part on to the table.

Let be linear mass density of chain.

To pull the chain we have to do work against the weight of hanging part of chain.

Let at any instant length of hanging part be x.

Therefore the weight of hanging part of chain is,

The work done in pulling the chain by small distance *dx *is,

Total work done to pull the whole of hanging part of chain is,

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Give two illustrations for each of following.

(a) Positive work

(b) Negative work

(c) Zero work.

Positive work:

(i) Work done by gravity on a free falling body is positive.

(ii) Work done by applied force when taken vertically up against gravity is positive.

Negative work:

(i) Work done by friction force on a moving body is negative.

(ii) The work done by gravity on a body moving up is negative.

Zero work:

(i)Work done by electrostatic force of nucleus on electron revolving in a circular orbit is zero.

(ii) Work done by tension force in string on a stone whirled in a circle is zero.

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Derive an expression for potential energy stored in spring.

Consider, a massless spring attached with mass *m* at one end and the other end of spring be connected with a rigid wall.

When we pull the mass towards C, the restoring force directed towards A is set up in spring. Work has to be done against this restoring force in order to displace the mass and hence this work is stored in the form of P.E in the spring.

Let at any instant, mass m be at a distance x from A.

Restoring force at this instance is,

Therefore, to keep the mas in equilibrium, we have to apply the force F_{a} equal and opposite to -F_{r}.

If the mass is further displaced by , the amount of work done dW for this dispalcement by applied force is,

Total amount of work done in order to displace the mass from mean position of A to C is using applied force F_{a} is,

Work done by applied force against this restoring force is stored in the form of potential energy is,

When we pull the mass towards C, the restoring force directed towards A is set up in spring. Work has to be done against this restoring force in order to displace the mass and hence this work is stored in the form of P.E in the spring.

Let at any instant, mass m be at a distance x from A.

Restoring force at this instance is,

Therefore, to keep the mas in equilibrium, we have to apply the force F

If the mass is further displaced by , the amount of work done dW for this dispalcement by applied force is,

Total amount of work done in order to displace the mass from mean position of A to C is using applied force F

Work done by applied force against this restoring force is stored in the form of potential energy is,

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Show that gravitational force is a conservative force.

A force is said to be conservative if work done by the force is independent of the path followed and depends upon the initial and final positions.

Suppose a body of mass m be taken from A to B along different paths as shown in the figure.

i) Work done if body is taken up straight along Ab,

W_{1} = -mgh

ii) Work done by gravity if body is taken up along path ACB,

W_{2} = W_{AC} + W_{CB}

iii) Work done by the gravity along the path ADEFGHB,

W_{3} = W_{AD} + W_{DE} + W_{EF} + W_{FG} + W_{GH} + W_{HB}_{ }

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Show that during a free fall - total energy (Kinetic + Potential) remains constant.

Let a body of mass m be dropped from point A at a height h from the ground.

At point A

Potential energy is,

∴ Total energy at point A,

...(1)

At point B

Let

Kinetic energy at B is,

Potential energy is,

∴ Total energy at point B is,

...(2)

**At ground:**

On reaching the ground, let velocity of the body be *V.*

∴ Total energy at point B is,

...(2)

∴

Kinetic energy at ground is,

Potential energy is,

∴ Total energy at point ground is,

Since total energy at A = total energy at B + total energy at ground.

Therefore total energy during free fall remains conserved.

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