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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 Îy near the Earthâs 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 ÎU = mg Îy. 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 ïr (1) ïU ï½ ïGM E m ï§ ï ï· ï½ 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 = Îy rirf â RE2 initial and final positions of the particle are close to the Earthâs surface: Substitute these expressions into Equation (1): ïU ï» GM E m ïy ï½ mg ïy 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 Earthâs atmosphere at which the âsurface

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