# 7.1 Work  (Page 7/11)

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## Work done by a spring force

A perfectly elastic spring requires 0.54 J of work to stretch 6 cm from its equilibrium position, as in [link] (b). (a) What is its spring constant k ? (b) How much work is required to stretch it an additional 6 cm?

## Strategy

Work “required” means work done against the spring force, which is the negative of the work in [link] , that is

$W=\frac{1}{2}k\left({x}_{B}^{2}-{x}_{A}^{2}\right).$

For part (a), ${x}_{A}=0$ and ${x}_{B}=6\text{cm}$ ; for part (b), ${x}_{B}=6\text{cm}$ and ${x}_{B}=12\text{cm}$ . In part (a), the work is given and you can solve for the spring constant; in part (b), you can use the value of k , from part (a), to solve for the work.

## Solution

1. $W=0.54\phantom{\rule{0.2em}{0ex}}\text{J}=\frac{1}{2}k\left[{\left(6\phantom{\rule{0.2em}{0ex}}\text{cm}\right)}^{2}-0\right]$ , so $k=3\phantom{\rule{0.2em}{0ex}}\text{N/cm}\text{.}$
2. $W=\frac{1}{2}\left(3\phantom{\rule{0.2em}{0ex}}\text{N/cm}\right)\left[{\left(12\phantom{\rule{0.2em}{0ex}}\text{cm}\right)}^{2}-{\left(6\phantom{\rule{0.2em}{0ex}}\text{cm}\right)}^{2}\right]=1.62\phantom{\rule{0.2em}{0ex}}\text{J}.$

## Significance

Since the work done by a spring force is independent of the path, you only needed to calculate the difference in the quantity $½k{x}^{2}$ at the end points. Notice that the work required to stretch the spring from 0 to 12 cm is four times that required to stretch it from 0 to 6 cm, because that work depends on the square of the amount of stretch from equilibrium, $½k{x}^{2}$ . In this circumstance, the work to stretch the spring from 0 to 12 cm is also equal to the work for a composite path from 0 to 6 cm followed by an additional stretch from 6 cm to 12 cm. Therefore, $4W\left(0\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}6\phantom{\rule{0.2em}{0ex}}\text{cm}\right)=W\left(0\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}6\phantom{\rule{0.2em}{0ex}}\text{cm}\right)+W\left(6\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}12\phantom{\rule{0.2em}{0ex}}\text{cm}\right)$ , or $W\left(6\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}12\phantom{\rule{0.2em}{0ex}}\text{cm}\right)=3W\left(0\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}6\phantom{\rule{0.2em}{0ex}}\text{cm}\right)$ , as we found above.

Check Your Understanding The spring in [link] is compressed 6 cm from its equilibrium length. (a) Does the spring force do positive or negative work and (b) what is the magnitude?

a. The spring force is the opposite direction to a compression (as it is for an extension), so the work it does is negative. b. The work done depends on the square of the displacement, which is the same for $x=±6\phantom{\rule{0.2em}{0ex}}\text{cm}$ , so the magnitude is 0.54 J.

## Summary

• The infinitesimal increment of work done by a force, acting over an infinitesimal displacement, is the dot product of the force and the displacement.
• The work done by a force, acting over a finite path, is the integral of the infinitesimal increments of work done along the path.
• The work done against a force is the negative of the work done by the force.
• The work done by a normal or frictional contact force must be determined in each particular case.
• The work done by the force of gravity, on an object near the surface of Earth, depends only on the weight of the object and the difference in height through which it moved.
• The work done by a spring force, acting from an initial position to a final position, depends only on the spring constant and the squares of those positions.

## Conceptual questions

Give an example of something we think of as work in everyday circumstances that is not work in the scientific sense. Is energy transferred or changed in form in your example? If so, explain how this is accomplished without doing work.

When you push on the wall, this “feels” like work; however, there is no displacement so there is no physical work. Energy is consumed, but no energy is transferred.

Give an example of a situation in which there is a force and a displacement, but the force does no work. Explain why it does no work.

#### Questions & Answers

definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
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position, velocity and acceleration of vector
Manuel Reply
hi
peter
hi
daniel
hi
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*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
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What is momentum
isijola
hello
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A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
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is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
same here please
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I think the question is that ,,, the working principal of eye and camera same or not?
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yes i think is same as the camera
muhammad
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
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Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
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Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
who is Newton?
John Reply
scientist
Jeevan
a scientist
Peter
that discovered law of motion
Peter
ok
John
but who is Isaac newton?
John
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
Daniel
and the color spectrum
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