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© Manhattan Press (H.K.) Ltd. 1
6.3 Gravitational potential 6.3 Gravitational potential energy and gravitational energy and gravitational
potentialpotential• • Gravitational potential energy (Gravitational potential energy (UU))• • Gravitational potential (Gravitational potential (VV))• • Potential gradientPotential gradient
© Manhattan Press (H.K.) Ltd. 2
Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (SB p. 215)
The gravitational potential energy (U) of a body of mass m at a point in a gravitational field is defined as the negative of work done by the gravitational force to bring the body from infinity to that point.
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More to Know 12More to Know 12
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Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (SB p. 215)
Object is moved dx towards earth
Work done (dW) = F dx =dx
x
mMG E
2
Note: Since the directions of gravitational force and the displacement are the same, the work done is positive.
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More to Know 13More to Know 13
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Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (SB p. 216)
Move object from x = to r
rmM
Gdxx
mMG
FdxWUEEr
r
2
)( PE nalGravitatio
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Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (SB p. 216)
1. –ve denotes U at is zero (highest) and decreases for closer to earth
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More to Know 14More to Know 14
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Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (SB p. 216)
2. Unit for U: joule (J)
3. r = RE:
h above surface:
Increase in U = U1 – Uo = mgh
4. Relationship between U and F
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More to Know 15More to Know 15
E
ER
mMGU o
)( 1 hR
mMGU
E
E
r FdrU
drdUF or
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Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (SB p. 217)
The gravitational potential (V) at a point in a gravitational field is the work done by the gravitational force to bring a unit mass from infinity to that point.
rGM
mUV E
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Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
1. V - at is zero- scalar- unit: J kg-1
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Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
2. VPQ – work done by gravitational force in bringing a unit mass from P to Q (independent of path)
21 rGM
rGM
VVVV EEQPPQ
Note: Since the directions of force and displacement is opposite in this definition, the work done (per unit mass) is negative.
© Manhattan Press (H.K.) Ltd. 10
Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
3. All points at same distance from earth’s centre have same V
The surface where all points on it has the same gravitational potential is known as an equipotential surface.
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Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
- No work done if move object on the same equipotential surface
- Gravitational field equipotential surfaces
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More to Know 16More to Know 16
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Example 6Example 6
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Potential gradient
6.3 Gravitational potential energy and gravitational potential (SB p. 220)
U = Work done = F r
m V = -Fg r = -mg r
g = - V/r
r 0
drdVg
potential gradient
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Potential gradient
6.3 Gravitational potential energy and gravitational potential (SB p. 220)
Relationship between V, g and r
Note: The gravitational field strength (g) is actually a vector and its value should be negative to represent its direction. In section 6.2, g is positive since we consider its magnitude only.
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Example 7Example 7
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Sign of work
The sign of work depends on the direction of force and displacement.
1. If their directions are the same, then positive work is done.
2. If their directions are opposite, then negative work is done. Return to
TextText
6.3 Gravitational potential energy and gravitational potential (SB p. 215)
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Other definitions of U
1. The work done by an external force to bring the body from infinity to that point.
2. The work done by the gravitational force to bring the body from that point to infinity.
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TextText
6.3 Gravitational potential energy and gravitational potential (SB p. 215)
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Sign of gravitational potential energy
The gravitational potential energy (U) of two particles at infinite separation is defined as zero by convention. Hence, U must be negative or zero (at infinity).
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TextText
6.3 Gravitational potential energy and gravitational potential (SB p. 216)
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r
r
FdrU
FdrUas written be alsocan
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TextText
6.3 Gravitational potential energy and gravitational potential (SB p. 217)
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VPQ can also be defined as the work done by an external force in bringing a unit mass from Q to P.
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TextText
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
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Equipotential surfaces around the earth
The equipotential surfaces around the earth are imaginary spherical shells with the same centre at the earth's centre.
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TextText
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
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Q:Q: The dashed lines in the figure represent the equipotential lines around the earth. The gravitational potential is as shown for each of the equipotential lines.
6.3 Gravitational potential energy and gravitational potential (SB p. 219)
© Manhattan Press (H.K.) Ltd. 22
Q:Q: (a) (i) Which one of the points (or points) has the highest gravitational potential? Explain your answer.
(ii) Calculate the work done by the gravitational field in bringing a spacecraft of mass 5 000 kg (1) from A to C; (2) from C to D.(b) (i) The equipotential lines, which are given
every 0.5 × 107 J kg−1, are not equally spaced. Explain why.
(ii) Calculate the distances AB and BC.( G = 6.7 × 10−11 N kg−2 m2; mass of earth = 6.0 × 1024 kg)
Solution
6.3 Gravitational potential energy and gravitational potential (SB p. 219)
© Manhattan Press (H.K.) Ltd. 23
Solution:Solution:
(a) (i) The point A has the highest gravitational potential.
Gravitational potential (V) =
Since the distance of A from the earth is the greatest, the value is the least negative or the highest.
(ii) (1) Work done by gravitational field to bring spacecraft from A to C:
= m ( VA − VC) = 5 000 [−4.0 ×107 − (−5.0 ×107)] = 5.0 ×1010 J
(2) Work done by gravitational field to bring spacecraft from C to D:
= m ( VC − VD) = 0 (for VC = VD)
6.3 Gravitational potential energy and gravitational potential (SB p. 219)
rGM E
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Solution (cont’d):Solution (cont’d):
(b) (i) The equipotential lines are not equally spaced because the gravitational potential does not vary linearly with r but varies inversely with r.
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TextText
6.3 Gravitational potential energy and gravitational potential (SB p. 219)
m 10908 m, 10071
m 100481005
10061076
m 109381054
10061076 m; 10001
1004
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VGM
r,r
GMV
rGM
V
CBBA
C
BA
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EA
A
EA
E
© Manhattan Press (H.K.) Ltd. 25
Q:Q: (a) Explain what is meant by(i) gravitational field strength, and(ii) gravitational potential.
Give an expression for each of these physical quantities and an equation relating the two quantities.(b) Show that the values of the gravitational field strength and the gravitational potential at any point of the earth’s surface are g and gRE respectively.Assume that the earth is a uniform surface of radius RE; and g is the acceleration of free fall on the earth’s surface.
6.3 Gravitational potential energy and gravitational potential (SB p. 221)
Solution
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Solution:Solution:
(a) (i) The gravitational field strength at a point in a gravitational field is the gravitational force acted on a unit mass at that point.
(ii) The gravitational potential (V) at a point in a gravitational field is the work done by the gravitational force to bring a unit mass from infinity to that point.
6.3 Gravitational potential energy and gravitational potential (SB p. 221)
2strength field nalGravitatio
r
GM
gradient) (Potentialstrength field onalGreavitati
)( potential nalGravitatio
drdVr
GMV
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Solution (cont’d) :Solution (cont’d) :
Return to
TextText
6.3 Gravitational potential energy and gravitational potential (SB p. 221)
EE
E
E
E
E
E
E
E
EE
E
gRRgR
RGM
V
g
gR
GM
mgmgR
mGMm
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GM
2
2
2
2
:surface searth' on the potential nalGravitatio . is surface searth' on thestrength field nalGravitatio
weight force nalgravitatio
surface, searth' on the mass ofobject an For
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:surface searth' On the (b)