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PhysicsDifficulty 2-3

Energy

The capacity to do work, taking forms such as kinetic energy (associated with motion) and potential energy (associated with position or configuration, such as height in a gravitational field). The total energy of an isolated system is conserved: energy changes form but is never created or destroyed.

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Work as an Integral of Force

W = Fd cos θ assumes a constant force. When a force varies as an object moves — such as the gravitational force between two masses whose separation changes across an appreciable range — the work done must instead be found by integrating force over the path.

W=∫xaxbF⋅dxW = \int_{x_a}^{x_b} \mathbf{F}\cdot d\mathbf{x}

Work done by a force as it moves an object from position xₐ to x_b, for a possibly varying force.

For a conservative force such as gravity, this integral defines a potential energy function U such that F = −dU/dx. Near Earth's surface, where the gravitational force on a mass m is the constant F = mg, integrating from 0 to height h recovers the familiar introductory result.

W=∫0hmg dx=mgh=ΔEpW = \int_0^h mg\, dx = mgh = \Delta E_p

E_p = mgh recovered as the special case of the general work integral for a constant force.

Info: This is exactly the same reasoning used to derive the gravitational potential energy of two widely separated masses, U(r) = −Gm₁m₂/r, by integrating Newton's law of gravitation instead of the constant near-surface approximation F = mg.

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Sources and methodology

  • The kinetic energy of a moving object is given by Eₖ = ½mv².
    • Fundamentals of Physics (Halliday, Resnick & Walker)
  • Gravitational potential energy near Earth's surface is given by Eₚ = mgh.
    • Fundamentals of Physics (Halliday, Resnick & Walker)
  • The law of conservation of energy states that the total energy of an isolated system remains constant — energy changes form but is never created or destroyed.
    • Fundamentals of Physics (Halliday, Resnick & Walker)

Content status: published, last reviewed 30 September 2026.