183
8-1 POTE NTIAL E N E RGY
Checkpoint 2
A particle is to move along an x axis from x ϭ 0 to x 1 while a conservative force, directed along the x axis, acts on the particle. The figure
shows three situations in which the x component of that force varies
with x.The force has the same maximum magnitude F 1 in all three situations. Rank the situations according to the change in the associated
potential energy during the particle’s motion, most positive first.
F 1
F 1
–F 1
x 1
x 1
x 1
(1)
( 2)
( 3)
the ground, (3) at the limb, and (4) 1.0 m above the limb?
Take the gravitational potential energy to be zero at y ϭ 0.
KEY IDEA
Once we have chosen the reference point for y ϭ 0, we can
calculate the gravitational potential energy U of the system
relative to that reference point with Eq. 8-9.
Calculations: For choice (1) the sloth is at y ϭ 5.0 m, and
U ϭ mgy ϭ (2.0 kg)(9.8 m/s
2
)(5.0 m)
ϭ 98 J.
(Answer)
For the other choices, the values of U are
(2) U ϭ mgy ϭ mg(2.0 m) ϭ 39 J,
(3) U ϭ mgy ϭ mg(0) ϭ 0 J,
(4) U ϭ mgy ϭ mg(Ϫ1.0 m)
ϭ Ϫ19.6 J Ϸ Ϫ20 J.
(Answer)
(b) The sloth drops to the ground. For each choice of reference point, what is the change ⌬U in the potential energy of
the sloth – Earth system due to the fall?
KEY IDEA
The change in potential energy does not depend on the
choice of the reference point for y ϭ 0; instead, it depends
on the change in height ⌬y.
Calculation: For all four situations, we have the same ⌬y ϭ
Ϫ5.0 m. Thus, for (1) to (4), Eq. 8-7 tells us that
⌬U ϭ mg ⌬y ϭ (2.0 kg)(9.8 m/s
2
)(Ϫ5.0 m)
ϭ Ϫ98 J.
(Answer)
Sample Problem 8.02 Choosing reference level for gravitational potential energy, sloth
Here is an example with this lesson plan: Generally you can
choose any level to be the reference level, but once chosen,
be consistent. A 2.0 kg sloth hangs 5.0 m above the ground
(Fig. 8-6).
(a) What is the gravitational potential energy U of the
sloth – Earth system if we take the reference point y ϭ 0 to be
(1) at the ground, (2) at a balcony floor that is 3.0 m above
Additional examples, video, and practice available at WileyPLUS
0
–3
–5
–6
3
0
–2
–3
5
2
0
6
3
1
0
(1)
(2)
(3)
(4)
Figure 8-6 Four choices of reference point y ϭ 0. Each y axis is marked
in units of meters. The choice affects the value of the potential energy
U of the sloth – Earth system. However, it does not affect the change
⌬U in potential energy of the system if the sloth moves by, say, falling.
becomes
which gives us
(elastic potential energy).
(8-11)
U(x) ϭ
1
2 kx
2
U Ϫ 0 ϭ
1
2 kx
2 Ϫ 0,
8-1 POTE NTIAL E N E RGY
Checkpoint 2
A particle is to move along an x axis from x ϭ 0 to x 1 while a conservative force, directed along the x axis, acts on the particle. The figure
shows three situations in which the x component of that force varies
with x.The force has the same maximum magnitude F 1 in all three situations. Rank the situations according to the change in the associated
potential energy during the particle’s motion, most positive first.
F 1
F 1
–F 1
x 1
x 1
x 1
(1)
( 2)
( 3)
the ground, (3) at the limb, and (4) 1.0 m above the limb?
Take the gravitational potential energy to be zero at y ϭ 0.
KEY IDEA
Once we have chosen the reference point for y ϭ 0, we can
calculate the gravitational potential energy U of the system
relative to that reference point with Eq. 8-9.
Calculations: For choice (1) the sloth is at y ϭ 5.0 m, and
U ϭ mgy ϭ (2.0 kg)(9.8 m/s
2
)(5.0 m)
ϭ 98 J.
(Answer)
For the other choices, the values of U are
(2) U ϭ mgy ϭ mg(2.0 m) ϭ 39 J,
(3) U ϭ mgy ϭ mg(0) ϭ 0 J,
(4) U ϭ mgy ϭ mg(Ϫ1.0 m)
ϭ Ϫ19.6 J Ϸ Ϫ20 J.
(Answer)
(b) The sloth drops to the ground. For each choice of reference point, what is the change ⌬U in the potential energy of
the sloth – Earth system due to the fall?
KEY IDEA
The change in potential energy does not depend on the
choice of the reference point for y ϭ 0; instead, it depends
on the change in height ⌬y.
Calculation: For all four situations, we have the same ⌬y ϭ
Ϫ5.0 m. Thus, for (1) to (4), Eq. 8-7 tells us that
⌬U ϭ mg ⌬y ϭ (2.0 kg)(9.8 m/s
2
)(Ϫ5.0 m)
ϭ Ϫ98 J.
(Answer)
Sample Problem 8.02 Choosing reference level for gravitational potential energy, sloth
Here is an example with this lesson plan: Generally you can
choose any level to be the reference level, but once chosen,
be consistent. A 2.0 kg sloth hangs 5.0 m above the ground
(Fig. 8-6).
(a) What is the gravitational potential energy U of the
sloth – Earth system if we take the reference point y ϭ 0 to be
(1) at the ground, (2) at a balcony floor that is 3.0 m above
Additional examples, video, and practice available at WileyPLUS
0
–3
–5
–6
3
0
–2
–3
5
2
0
6
3
1
0
(1)
(2)
(3)
(4)
Figure 8-6 Four choices of reference point y ϭ 0. Each y axis is marked
in units of meters. The choice affects the value of the potential energy
U of the sloth – Earth system. However, it does not affect the change
⌬U in potential energy of the system if the sloth moves by, say, falling.
becomes
which gives us
(elastic potential energy).
(8-11)
U(x) ϭ
1
2 kx
2
U Ϫ 0 ϭ
1
2 kx
2 Ϫ 0,