Gravitational potential energy
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Lecture 14: Gravitational potential energy and space travel • Universal gravitational potential energy • Space travel problems • Escape speed • Orbital energy • Multiple objects Gravitational potential energy πΉππππ£ = πΊππ , π2 attractive Conservative force ππ΅ ππ΅ − ππ΄ = −ππ΄→π΅ = − πΉ β ππ ππ΄ ππ΅ ππ΅ − ππ΄ = −ππ΄→π΅ = − πΉ β ππ ππ΄ ππ΅ ππ΅ πΉ β ππ = ππ΄ ππ΅ πΉπ ππ = ππ΄ πΊππ =− − π ππ΄ ππ΅ ππ΄ πΊππ − 2 ππ π πΊππ πΊππ = − ππ΅ ππ΄ 1 1 ππ΅ − ππ΄ = −πΊππ − ππ΅ ππ΄ Choose reference point: π0 = ∞. Assign π π0 = ∞ = 0 ππΊπππ£ πΊππ =− π negative! Potential energy diagram π π0 = ∞ = 0 Potential energy diagram Escape condition πΈ= ½ππ£ 2 πΊππ − π Critical escape condition: barely making it to π = πππππππ‘π¦, having slowed to speed of zero Escape speed Minimum speed an object must have at distance R from central mass M if it is to go infinitely far away πΈπ = πΈπ 2 ½ππ£ππ π πΊππ πΊππ 2 − = ½ππ£∞ − π π=∞ 2 ½ππ£ππ π πΊππ − =0−0 π π£ππ π = 2πΊπ π Example: Escape speed from Earth ππΈπππ‘β = 5.97 × 1024 ππ π πΈπππ‘β = 6.38 × 106 π πΊ = 6.67 × 10−11 π π2/ππ2 π£ππ π = 2πΊπ = 11,200 π/π π Example: Escape speed from orbit Speed necessary to escape gravitational field of the Sun when object is launched from Earth: πππ’π=1.99 × 10 30 kg π ππΈ = 1.50 × 10 11 m πΊ= 6.67 × 10 -11 N m2/kg2 π£ππ π = 2πΊπππ’π = 42,000 π/π π ππΈ Orbital Energy πΈ =πΎ+π = ½ππ£ 2 πΊππ − π Speed of satellite in πΊπ circular orbit: π£ 2 = π πΊπ πΊππ πΈ = ½π − π π πΊππ πΈ=− 2π Satellite Motion πΉπ₯ = πΉππππ£,π₯ = ππ ππ‘ ππ₯ =ππ ππ‘ (+ππΆ ) πΊππ ππ‘ π π£2 = ππ ππ‘ 2 π π πΊπ = π£2 π Multiple objects Example with multiple objects A planet has mass 4π and a radius 2π. Its moon has mass π and radius π. The centers of the planet and moon are a distance 9π apart. A shuttle of mass π is a distance 4π away from the center of the planet and moving with speed π. What is the total mechanical energy of the shuttle? V 4M 2r 4r m 9r M r If the shuttle was initially at rest at X, how much work did the engines do? V 4M 2r 4r m 9r M r
