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```because the speed Ï is proportional to the square root of the mass of the planet, the
speed increases as the mass of the planet increases.
Example 13.6 The Change in Potential Energy
A particle of mass m is displaced through a small vertical distance Îy near the Earthâs
surface. Show that in this situation the general expression for the change in
gravitational potential energy given by Equation 13.13 reduces to the familiar
relationship ÎU = mg Îy.
SOLUTION
Conceptualize Compare the two different situations for which we have developed
expressions for gravitational potential energy: (1) a planet and an object that are far
apart for which the energy expression is Equation 13.14 and (2) a small object at the
surface of a planet for which the energy expression is Equation 7.19. We wish to
show that these two expressions are equivalent.
Categorize This example is a substitution problem.
Combine the fractions in Equation
13.13:
ï¦ 1 1ï¶
ï¦ r ï­r
(1) ïU ï½ ï­GM E m ï§ ï­ ï· ï½ GM E m ï§ f i
ï§r r ï·
ï§ rr
i ï¸
ï¨ f
ï¨ i f
ï¶
ï·ï·
ï¸
Evaluate rf â ri and rirf if both the rf â ri = Îy rirf â RE2
initial and final positions of the
particle are close to the Earthâs
surface:
Substitute these expressions into
Equation (1):
ïU ï»
GM E m
ïy ï½ mg ïy
RE 2
where g = GME/RE2 (Eq. 13.5).
WHAT IF? Suppose you are performing upper-atmosphere studies and are asked by
your supervisor to find the height in the Earthâs atmosphere at which the âsurface
```
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