1© Manhattan Press (H.K.) Ltd. 6.3 Gravitational potential energy and gravitational potential Gravitational potential energy (U) Gravitational potential

© 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


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




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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Move object from x = to r

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mMG

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2

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1. –ve denotes U at is zero (highest) and decreases for closer to earth

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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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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.

rGM

mUV E




1. V - at is zero- scalar- unit: J kg-1




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.




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.




- No work done if move object on the same equipotential surface

- Gravitational field equipotential surfaces

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Example 6Example 6


Potential gradient


U = Work done = F r

m V = -Fg r = -mg r

g = - V/r

r 0

drdVg

potential gradient


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 7Example 7


End


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

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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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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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r

r

FdrU

FdrUas written be alsocan

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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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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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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.



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



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)


rGM E


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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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.


Solution


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.


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drdVr

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Solution (cont’d) :Solution (cont’d) :

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1© Manhattan Press (H.K.) Ltd. 6.3 Gravitational potential energy and gravitational potential Gravitational potential energy (U) Gravitational potential