6 3 Gravitational potential energy and gravitational potential

6. 3 Gravitational potential energy and gravitational potential • Gravitational potential energy (U) • Gravitational potential (V) • Potential gradient © Manhattan Press (H. K. ) Ltd. 1

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. Go to More to Know 12 © Manhattan Press (H. K. ) Ltd. 2

6. 3 Gravitational potential energy and gravitational potential (SB p. 215) Gravitational potential energy (U) Object is moved dx towards earth Work done (d. W) = F dx = Note: Since the directions of gravitational force and the displacement are the same, the work done is positive. © Manhattan Press (H. K. ) Ltd. Go to More to Know 13 3

6. 3 Gravitational potential energy and gravitational potential (SB p. 216) Gravitational potential energy (U) Move object from x = to r © Manhattan Press (H. K. ) Ltd. 4

6. 3 Gravitational potential energy and gravitational potential (SB p. 216) Gravitational potential energy (U) 1. –ve denotes U at is zero (highest) and decreases for closer to earth Go to More to Know 14 © Manhattan Press (H. K. ) Ltd. 5

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 = U 1 – Uo = mgh 4. Relationship between U and F Go to More to Know 15 © Manhattan Press (H. K. ) Ltd. 6

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. © Manhattan Press (H. K. ) Ltd. 7

6. 3 Gravitational potential energy and gravitational potential (SB p. 218) Gravitational potential (V) 1. V - at is zero - scalar - unit: J kg-1 © Manhattan Press (H. K. ) Ltd. 8

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. © Manhattan Press (H. K. ) Ltd. 9

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. © Manhattan Press (H. K. ) Ltd. 10

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 Go to More to Know 16 Go to More to Know 17 Go to © Manhattan Press (H. K. ) Ltd. Example 6 11

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 © Manhattan Press (H. K. ) Ltd. 12

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. Go to © Manhattan Press (H. K. ) Ltd. Example 7 13

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6. 3 Gravitational potential energy and gravitational potential (SB p. 215) 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 © Manhattan Press (H. K. ) Ltd. Text 15

6. 3 Gravitational potential energy and gravitational potential (SB p. 215) 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. Return to © Manhattan Press (H. K. ) Ltd. Text 16

6. 3 Gravitational potential energy and gravitational potential (SB p. 216) 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). Return to © Manhattan Press (H. K. ) Ltd. Text 17

6. 3 Gravitational potential energy and gravitational potential (SB p. 217) Return to © Manhattan Press (H. K. ) Ltd. Text 18

6. 3 Gravitational potential energy and gravitational potential (SB p. 218) VPQ can also be defined as the work done by an external force in bringing a unit mass from Q to P. Return to © Manhattan Press (H. K. ) Ltd. Text 19

6. 3 Gravitational potential energy and gravitational potential (SB p. 218) Equipotential surfaces around the earth The equipotential surfaces around the earth are imaginary spherical shells with the same centre at the earth's centre. Return to © Manhattan Press (H. K. ) Ltd. Text 20

6. 3 Gravitational potential energy and gravitational potential (SB p. 219) 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. © Manhattan Press (H. K. ) Ltd. 21

6. 3 Gravitational potential energy and gravitational potential (SB p. 219) 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 m 2; mass of earth = 6. 0 × 1024 kg) Solution © Manhattan Press (H. K. ) Ltd. 22

6. 3 Gravitational potential energy and gravitational potential (SB p. 219) 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 ( V C − V D) = 0 © Manhattan Press (H. K. ) Ltd. (for VC = VD) 23

6. 3 Gravitational potential energy and gravitational potential (SB p. 219) 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. Return to © Manhattan Press (H. K. ) Ltd. Text 24

6. 3 Gravitational potential energy and gravitational potential (SB p. 221) 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 g. RE 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 © Manhattan Press (H. K. ) Ltd. 25

6. 3 Gravitational potential energy and gravitational potential (SB p. 221) 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. © Manhattan Press (H. K. ) Ltd. 26

6. 3 Gravitational potential energy and gravitational potential (SB p. 221) Solution (cont’d) : Return to Text © Manhattan Press (H. K. ) Ltd. 27