1. 6.3 Gravitational potential energy and gravitational potential
• Gravitational potential energy (U)
• Gravitational potential (V)
• Potential gradient
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2. Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (SB p. 215)
Gravitational potential energy (U)
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 12
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3. Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (SB p. 215)
Gravitational potential energy (U)
Object is moved dx towards earth
Work done (dW) = F dx =
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 13
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4. Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (SB p. 216)
Gravitational potential energy (U)
Move object from x =  to r
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5. Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (SB p. 216)
Gravitational potential energy (U)
–ve denotes U at  is zero (highest) and decreases for closer to earth
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More to Know 14
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6. Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (SB p. 216)
Gravitational potential energy (U)
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 15
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7. Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (SB p. 217)
Gravitational potential (V)
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.
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8. Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
Gravitational potential (V)
V - at  is zero - scalar - unit: J kg-1
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9. Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
Gravitational potential (V)
2. VPQ – work done by gravitational force in bringing a unit mass from P to Q (independent of path)
Note: Since the directions of force and displacement is opposite in this definition, the work done (per unit mass) is negative.
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10. Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
Gravitational potential (V)
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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11. Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
Gravitational potential (V)
- No work done if move object on the same equipotential surface
- Gravitational field  equipotential surfaces
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More to Know 16
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More to Know 17
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Example 6
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12. Potential gradient U = Work done = F r m V = -Fg r = -mg r
6.3 Gravitational potential energy and gravitational potential (SB p. 220)
Potential gradient
U = Work done = F r
m V = -Fg r = -mg r
g = - V/r
r  0
potential gradient
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13. Potential gradient Relationship between V, g and r Example 7
6.3 Gravitational potential energy and gravitational potential (SB p. 220)
Potential gradient
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 7
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14. End
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15. More to Know 12 Text Sign of work
6.3 Gravitational potential energy and gravitational potential (SB p. 215)
More to Know 12
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.
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16. More to Know 13 Text Other definitions of U
6.3 Gravitational potential energy and gravitational potential (SB p. 215)
More to Know 13
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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17. More to Know 14 Text Sign of gravitational potential energy
6.3 Gravitational potential energy and gravitational potential (SB p. 216)
More to Know 14
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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18. More to Know 15 Text Return to
6.3 Gravitational potential energy and gravitational potential (SB p. 217)
More to Know 15
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19. 6. 3 Gravitational potential energy and gravitational potential (SB p
More to Know 16
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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Text
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20. More to Know 17 Text Equipotential surfaces around the earth
6.3 Gravitational potential energy and gravitational potential (SB p. 218)
More to Know 17
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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21. 6. 3 Gravitational potential energy and gravitational potential (SB p
Example 6
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.
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22. 6. 3 Gravitational potential energy and gravitational potential (SB p
Example 6 (cont'd)
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 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
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23. 6. 3 Gravitational potential energy and gravitational potential (SB p
Example 6
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) = [−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)
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24. Example 6 Text Solution (cont’d):
6.3 Gravitational potential energy and gravitational potential (SB p. 219)
Example 6
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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25. 6. 3 Gravitational potential energy and gravitational potential (SB p
Example 7
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.
Solution
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26. 6. 3 Gravitational potential energy and gravitational potential (SB p
Example 7
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.
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27. Example 7 Text Solution (cont’d) : Return to
6.3 Gravitational potential energy and gravitational potential (SB p. 221)
Example 7
Solution (cont’d) :
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