Work Done by Varying Force

8/2/2019 Work Done by Varying Force

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Physics

Work done by varying Force

A: Learning Goals

Work done by a varying force

Work done by a spring

B : Prerequisites

Work done by a constant force

Vectors and their representation in the Cartesian coordinate system

Scalar product of two vectors

Free-body diagrams

Basic knowledge of kinematics and laws of motion

Differentiating simple polynomial and trigonometric functions

Integration of simple polynomial and trigonometric functions

C: Work done by a Varying Force

Let us consider a particle being displaced along the x axis under the action of a varying force. The

particle is displaced in the direction of increasing x from x = xi to x = xf . In such a situation, we cannot

use W = (F cos ).d to calculate the work done by the force because this relationship applies only

when force F is constant in magnitude and direction. However, if we imagine that the particle

undergoes a very small displacement , as shown in figure, then the x-component of force is

approximately constant over this interval; for this small displacement, we can express the work done

by the force as . This is just the area of the shaded rectangle in Figure shown here.

If we express the resultant force in the x

direction as , then work done as the

particle moves from to will be:



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If we imagine that versus curve is divided into a large number of such intervals, then the

total work done for the displacement from x = xi to x = xf is approximately equal to the sum of a

large number of such terms. That is:

∑ .

The work done by the force component

for the small displacement is ,

which equals the area of the shaded

rectangle. The total work done for the

displacement from x = xi to x = xf isapproximately equal to the sum of the

areas of all the rectangles.

The work done by the component of the varying force as the

particle moves from x = xi to x = xf is

exactly equal to the area under this

curve. We can express the work

done by as the particle moves

from to as:

If more than one force acts on aparticle, the total work done is just

the work done by the resultant

force. If we express the resultant

force in the x- direction as ∑ ;

then the total work, or net work

done as the particle moves from

to is:



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If the displacements are allowed to approach zero, then the number of terms in the sum increases

without limit but the value of the sum approaches a definite value equal to the area bounded by the

curve and the x axis:

This definite integral is numerically equal to the area under the versus curve between and . Therefore, we can express the work done by as the particle moves from to as:

If more than one force acts on a particle, the total work done is just the work done by the resultant

force. If we express the resultant force in the x- direction as ∑ ;

then the total work, or net work done as the particle moves from to is:

Heading: Work done by spring



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A common physical system for which the force varies with position is shown here. A block

on a horizontal, frictionless surface is connected to a spring. If the spring is either elongated

(stretched) or compressed a small distance from its unstretched (equilibrium) configuration,

it exerts on the block a force of magnitude;

Where x is the displacement of the block from its unstretched ( ) position and is a

positive constant called the force constant or stiffness of the spring.

The negative sign signifies that the force exerted by the spring is always directed opposite

the displacement.

In other words, the force required to stretch or compress a spring is proportional to the

amount of elongation or compression; . This force law for springs is known as Hooke’s law.

Stiff springs have large values, and soft springs have small values.

x max



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If the spring is compressed until the block is at the point and is then released, the

block moves from through zero to .

And, if the spring is instead stretched until the block is at the point

and is then

released, the block moves from through zero to . It then reverses direction,

returns to , and continues oscillating back and forth.

Heading: Spring is unstretched

When spring is in normal position or unstretched i.e. as shown here. In this position,

force exerted by the spring ; because the spring force always acts toward the

equilibrium position , this force is also known as the restoring force.

When (natural length of the spring), the spring force is directed to the left and known as

restoring force. In this position, force exerted by the spring .

- x max



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Heading: Spring is compressed

Let us now suppose the block has been pushed to the left a distance xmax from equilibrium

and is then released. Let us calculate the work, Ws done by the spring force as the block

moves from to . Then, work done by the spring is:

()

Heading: Spring is stretched

()

When x is negative (compressed spring), the spring force is directed to the right, in the positive x-

direction. Work done by the spring force as the block moves from to is:



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Now the block has been pushed to the right a distance xmax from equilibrium and is then

released. Let us calculate the work, Ws done by the spring force as the block moves from to . Then, work done by the spring is:

()

Therefore, the net work done by the spring force as the block moves from to

is zero.

Heading: Graph of versus



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If we plot a graph of versus for the block-spring system then the graph will be a straight line as

shown here. The work done by the spring force as the block moves from to 0 (zero) is the

area of the shaded triangle, .

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Work Done by Varying Force