190 Potential Energy Formula Derivation
A derivation of the potential energy formula that is used to obtain \(U(x_{2})-U(x_{1})=-\displaystyle\int_{x_{1}}^{x_{2}} F(x)\, dx\) from the work-energy theorem.
Related definitions:
The work-energy theorem is
\[ K(t_{2}) - K(t_{1}) = \displaystyle\int_{x_{1}}^{x_{2}} F(x)\, dx \]
The potential-energy change is
\[ U(x_{2}) - U(x_{1}) = -\displaystyle\int_{x_{1}}^{x_{2}} F(x)\, dx \]
where
- \(K\) is the kinetic energy.
- \(t_{1}\) and \(t_{2}\) are the initial and final times.
- \(F(x)\) is a position-dependent force.
- \(x_{1}\) and \(x_{2}\) are the initial and final positions.
- \(U(x)\) is the potential energy.
190.1 References
- Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — from \(\dfrac{dK}{dt} = F\,\dfrac{dx}{dt}\) to \(U(x_{2})-U(x_{1})=-\int F\,dx\).
- Annihilation
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- Potential Energy Formula Derivation
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