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AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 3: Work, Energy & PowerTranslational Kinetic Energy & Potential EnergyGravitational Potential Energy Between Objects

Gravitational Potential Energy Between Objects (College Board AP® Physics 1: Algebra-Based)

Study Guide

Leander Oates

Written by: Leander Oates

Reviewed by: Caroline Carroll

Gravitational potential energy between objects

Gravitational potential energy between two objects

  • When a system consists of two approximately spherical masses such as moons, planets, or stars, the absolute gravitational potential energy of the system is given by:

U subscript g space equals space minus G fraction numerator m subscript 1 m subscript 2 over denominator r end fraction

  • Where:

    • U subscript g = absolute gravitational potential energy, measured in straight J

    • G = Universal gravitational constant open parentheses 6.67 cross times 10 to the power of negative 11 end exponent space straight N times straight m squared divided by kg squared close parentheses

    • m subscript 1 m subscript 2 = the masses of the two objects, measured in kg

    • r = the separation distance between the center of mass of each object, measured in straight m

  • Notice that the equation looks very similar to Newton's law of gravitation

open vertical bar F with rightwards arrow on top subscript g close vertical bar space equals space G fraction numerator m subscript 1 m subscript 2 over denominator r squared end fraction

  • Consider the masses to be Earth and the Moon

  • The gravitational pull of the Earth on the Moon is:

open vertical bar F with rightwards arrow on top subscript g close vertical bar space equals space m subscript M g subscript E

  • The gravitational field strength is no longer constant when the object is far away from the Earth's surface

  • Therefore, the absolute gravitational potential energy of the Moon is:

U subscript g space M end subscript space equals space m subscript M g subscript E increment y

  • Substituting in the derived expression for g subscript E:

U subscript g space M end subscript space equals space F subscript g increment y

  • Where increment y is the separation distance r between the center of mass of the Moon and the center of mass of Earth, which gives:

U subscript g space M end subscript space equals space open parentheses G fraction numerator m subscript 1 m subscript 2 over denominator r squared end fraction close parentheses r space equals space G fraction numerator m subscript E m subscript M over denominator r end fraction

  • The value of U subscript g is always negative

U subscript g space M end subscript space equals space minus G fraction numerator m subscript E m subscript M over denominator r end fraction

  • When the separation distance between the objects approaches infinity, the gravitational field strength tends toward zero

  • As an object moves away from the Earth, work is done against the gravitational field

  • The object's gravitational potential energy increases with distance from the Earth to a maximum value at an infinite distance

  • Therefore, the point of maximum gravitational potential energy is defined to be the zero point

  • This means that absolute gravitational potential energy will always have a negative value

Gravitational potential energy of a system with more than two objects

  • Gravitational potential energy is a scalar quantity

  • Therefore, when a system contains more than two objects, the gravitational potential energy of the system is the sum of the gravitational potential energies of each pair of objects

  • For a system consisting of a central object (e.g. a planet, moon, or star) and two satellites, the total gravitational potential energy of the system is:

U subscript g space s y s end subscript space equals space U subscript g space 1 end subscript space plus space U subscript g space 2 end subscript

  • Where

    • U subscript g space s y s end subscript = total gravitational potential energy of the system, measured in straight J

    • U subscript g space 1 end subscript = gravitational potential energy of the central object and satellite 1, measured in straight J

    • U subscript g space 2 end subscript = gravitational potential energy of the central object and satellite 2, measured in straight J

Worked Example

A satellite of mass 4.5 cross times 10 cubed space kg is transported into an orbit 2000 space km above the Earth's surface by a rocket of mass 8 cross times 10 cubed space kg. When the orbit is reached, the rocket releases the satellite.

The Earth has a mass of 6 cross times 10 to the power of 24 space kg and a radius of 6.4 cross times 10 to the power of 6 space straight m.

Calculate the total gravitational potential energy of the system just after separation.

Answer:

Step 1: Analyze the scenario and list the known quantities

  • Consider the system to be the Earth, the rocket, and the satellite

    • Mass of Earth, m subscript E space equals space 6 cross times 10 to the power of 24 space kg

    • Mass of rocket, m subscript R space equals space 8 cross times 10 cubed space kg

    • Mass of satellite, M subscript S space equals space 4.5 cross times 10 cubed space kg

    • Radius of Earth, r subscript E space equals space 6.4 cross times 10 to the power of 6 space straight m

    • Distance between Earth and satellite in orbit, d subscript S space equals space 2000 space km space equals space 2 cross times 10 to the power of 6 space straight m

Step 2: Write an expression for the total gravitational potential energy of the system

  • The absolute gravitational potential energy of the system just after separation is:

U subscript g space s y s end subscript space equals space U subscript g space R end subscript space plus space U subscript g space S end subscript

U subscript g space s y s end subscript space equals space open parentheses negative G fraction numerator m subscript E m subscript R over denominator r subscript E space plus space d subscript S end fraction close parentheses space plus space open parentheses negative G fraction numerator m subscript E m subscript S over denominator r subscript E space plus space d subscript S end fraction close parentheses

U subscript g space s y s end subscript space equals space minus G fraction numerator m subscript E over denominator r subscript E space plus space d subscript S end fraction open parentheses m subscript R space plus space m subscript S close parentheses

Step 3: Calculate the total gravitational potential energy of the system

U subscript g space s y s end subscript space equals space minus open parentheses 6.67 cross times 10 to the power of negative 11 end exponent close parentheses fraction numerator 6 cross times 10 to the power of 24 over denominator open parentheses 6.4 cross times 10 to the power of 6 space plus space 2 cross times 10 to the power of 6 close parentheses end fraction open parentheses 8 cross times 10 cubed space plus space 4.5 cross times 10 cubed close parentheses

U subscript g space s y s end subscript space equals space minus 5.96 cross times 10 to the power of 11 space straight J

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    Author
    Reviewer
    Leander Oates

    Author: Leander Oates

    Expertise: Physics

    Leander graduated with First-class honours in Science and Education from Sheffield Hallam University. She won the prestigious Lord Robert Winston Solomon Lipson Prize in recognition of her dedication to science and teaching excellence. After teaching and tutoring both science and maths students, Leander now brings this passion for helping young people reach their potential to her work at SME.

    Caroline Carroll

    Author: Caroline Carroll

    Expertise: Physics Subject Lead

    Caroline graduated from the University of Nottingham with a degree in Chemistry and Molecular Physics. She spent several years working as an Industrial Chemist in the automotive industry before retraining to teach. Caroline has over 12 years of experience teaching GCSE and A-level chemistry and physics. She is passionate about creating high-quality resources to help students achieve their full potential.