161
Caution: If the block is not stationary before and after the displacement, then this
statement is not true.
Work W s is positive if the block ends up closer to the relaxed position (x ϭ 0) than
it was initially. It is negative if the block ends up farther away from x ϭ 0. It is zero
if the block ends up at the same distance from x ϭ 0.
If x i ϭ 0 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 x axis while continuing to apply a
force to it. During the displacement, our applied force does work W a on the block
F
:
a
W s ϭ Ϫ
1
2 kx
2
7-4 WOR K DON E BY A SPR I NG 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 x axis for the
block in Fig. 7-10 are (a) Ϫ3 cm, 2 cm; (b) 2 cm, 3 cm; and (c) Ϫ2 cm, 2 cm. In each situation, 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 cannot find 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 displacement up into an infinite number of tiny parts and then approximate 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 m ϭ 0.40 kg slides
across a horizontal frictionless counter with speed v ϭ 0.50 m/s.
Multiplied out, this yields
(work by a spring force).
(7-25)
This work W s done by the spring force can have a positive or negative value,
depending on whether the net transfer of energy is to or from the block as the
block moves from x i to x f . Caution: The final position x f appears in the second
term on the right side of Eq. 7-25. Therefore, Eq. 7-25 tells us:
W s ϭ
1
2 kx i
2 Ϫ
1
2 kx f
2
Figure 7-11 A canister moves toward a spring.
k
m
Frictionless
First touch
Stop
v
d
The spring force does
negative work, decreasing
speed and kinetic energy.
while the spring force does work W s . By Eq. 7-10, the change ⌬K in the kinetic energy of the block due to these two energy transfers is
⌬K ϭ K f Ϫ K i ϭ W a ϩ W s ,
(7-27)
in which K f is the kinetic energy at the end of the displacement and K i is that at
the start of the displacement. If the block is stationary before and after the displacement, then K f and K i are both zero and Eq. 7-27 reduces to
W a ϭ ϪW s .
( 7 - 2 8 )
Caution: If the block is not stationary before and after the displacement, then this
statement is not true.
Work W s is positive if the block ends up closer to the relaxed position (x ϭ 0) than
it was initially. It is negative if the block ends up farther away from x ϭ 0. It is zero
if the block ends up at the same distance from x ϭ 0.
If x i ϭ 0 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 x axis while continuing to apply a
force to it. During the displacement, our applied force does work W a on the block
F
:
a
W s ϭ Ϫ
1
2 kx
2
7-4 WOR K DON E BY A SPR I NG 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 x axis for the
block in Fig. 7-10 are (a) Ϫ3 cm, 2 cm; (b) 2 cm, 3 cm; and (c) Ϫ2 cm, 2 cm. In each situation, 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 cannot find 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 displacement up into an infinite number of tiny parts and then approximate 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 m ϭ 0.40 kg slides
across a horizontal frictionless counter with speed v ϭ 0.50 m/s.
Multiplied out, this yields
(work by a spring force).
(7-25)
This work W s done by the spring force can have a positive or negative value,
depending on whether the net transfer of energy is to or from the block as the
block moves from x i to x f . Caution: The final position x f appears in the second
term on the right side of Eq. 7-25. Therefore, Eq. 7-25 tells us:
W s ϭ
1
2 kx i
2 Ϫ
1
2 kx f
2
Figure 7-11 A canister moves toward a spring.
k
m
Frictionless
First touch
Stop
v
d
The spring force does
negative work, decreasing
speed and kinetic energy.
while the spring force does work W s . By Eq. 7-10, the change ⌬K in the kinetic energy of the block due to these two energy transfers is
⌬K ϭ K f Ϫ K i ϭ W a ϩ W s ,
(7-27)
in which K f is the kinetic energy at the end of the displacement and K i is that at
the start of the displacement. If the block is stationary before and after the displacement, then K f and K i are both zero and Eq. 7-27 reduces to
W a ϭ ϪW s .
( 7 - 2 8 )