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# Work, Kinetic Energy and Potential Energy

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128
CHAPTER 6. WORK, KINETIC ENERGY AND POTENTIAL ENERGY
Work can be negative; this happens when the angle between force and displacement is
larger than 90
. It can also be zero; this happens if = 90
. To do work, the force must
have a component along (or opposite to) the direction of the motion.
If several different (constant) forces act on a mass while it moves though a displacement
d, then we can talk about the net work done by the forces,
W
net
= F
1
· d + F
1
· d + F
1
· d + . . .
(6.4)
=
F · d
(6.5)
= F
net
· d
(6.6)
If the force which acts on the object is not constant while the object moves then we must
perform an integral (a sum) to find the work done.
Suppose the object moves along a straight line (say, along the x axis, from x
i
to x
f
) while
a force whose x component is F
x
(x) acts on it. (That is, we know the force F
x
as a function
of x.) Then the work done is
W =
x
f
x
i
F
x
(x) dx
(6.7)
Finally, we can give the most general expression for the work done by a force. If an object
moves from r
i
= x
i
i + y
i
j + z
i
k to r
f
= x
f
i + y
f
j + z
f
k while a force F(r) acts on it the work
done is:
W =
x
f
x
i
F
x
(r) dx +
y
f
y
i
F
y
(r) dy +
z
f
z
i
F
z
(r) dz
(6.8)
where the integrals are calculated along the path of the object's motion. This expression
can be abbreviated as
W =
r
f
r
i
F · dr .
(6.9)
This is rather abstract! But most of the problems where we need to calculate the work done
by a force will just involve Eqs. 6.3 or 6.7
We're familiar with the force of gravity; gravity does work on objects which move ver-
tically. One can show that if the height of an object has changed by an amount y then
gravity has done an amount of work equal to
W
grav
= -mgy
(6.10)
regardless of the horizontal displacement. Note the minus sign here; if the object increases
in height it has moved oppositely to the force of gravity.
6.1.3
Spring Force
The most famous example of a force whose value depends on position is the spring force,
which describes the force exerted on an object by the end of an ideal spring. An ideal
spring will pull inward on the object attached to its end with a force proportional to the
amount by which it is stretched; it will push outward on the object attached to its with a
force proportional to amount by which it is compressed.

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Summary :

## If an object moves from r i = x i i + y i j + z i k to r f = x f i + y f j + z f k while a force F(r) acts on it the work done is: W = x f x i F x (r) dx + y f y i F y (r) dy + z f z i F z (r) dz (6.8) where the integrals are calculated along the path of the object's motion.

Tags : object,done,which,moes,graity,while,spring,amount,along,then,acts,displacement,net
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