9781118230725.pdf

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Caution:If the block is not stationary before and after the displacement, then this
statement is nottrue.

Work Wsis positive if the block ends up closer to the relaxed position (x0) than

it was initially. It is negative if the block ends up farther away from x0. It is zero

if the block ends up at the same distance from x0.

Ifxi0 and if we call the final position x, then Eq. 7-25 becomes

(work by a spring force). (7-26)

The Work Done by an Applied Force

Now suppose that we displace the block along the xaxis while continuing to apply a
force F to it. During the displacement, our applied force does work Waon the block
:
a

Ws^12 kx^2

7-4 WORK DONE BY A SPRING FORCE

If a block that is attached to a spring is stationary before and after a displacement,

then the work done on it by the applied force displacing it is the negative of the

work done on it by the spring force.

Checkpoint 2

For three situations, the initial and final positions, respectively, along the xaxis for the

block in Fig. 7-10 are (a) 3 cm, 2 cm; (b) 2 cm, 3 cm; and (c) 2 cm, 2 cm. In each sit-

uation, is the work done by the spring force on the block positive, negative, or zero?

Sample Problem 7.06 Work done by a spring to change kinetic energy

When a spring does work on an object, we cannotfind the

work by simply multiplying the spring force by the object’s

displacement. The reason is that there is no one value for

the force—it changes. However, we can split the displace-

ment up into an infinite number of tiny parts and then ap-

proximate the force in each as being constant. Integration

sums the work done in all those parts. Here we use the

generic result of the integration.

In Fig. 7-11, a cumin canister of mass m0.40 kg slides

across a horizontal frictionless counter with speed v0.50 m/s.

Multiplied out, this yields

(work by a spring force). (7-25)

This work Wsdone by the spring force can have a positive or negative value,
depending on whether the nettransfer of energy is to or from the block as the
block moves from xitoxf.Caution:The final position xfappears in the second
term on the right side of Eq. 7-25. Therefore, Eq. 7-25 tells us:

Ws^12 kxi^2 ^12 kxf^2

Figure 7-11A canister moves toward a spring.

k

Frictionless m

Stop First touch

v

d

The spring force does

negative work, decreasing

speed and kinetic energy.

while the spring force does work Ws. By Eq. 7-10, the change Kin the kinetic en-
ergy of the block due to these two energy transfers is

KKfKiWaWs, (7-27)

in which Kfis the kinetic energy at the end of the displacement and Kiis that at
the start of the displacement. If the block is stationary before and after the dis-
placement, then KfandKiare both zero and Eq. 7-27 reduces to

WaWs. (7-28)