Source: High school physics (Chinese)
Problem Sets:
Problem
The Earth can be treated as a point mass $M = 6.0 \times 10^{24}$ kg with radius $R_e = 6.4 \times 10^6$ m. An object has mass $m = 1$ kg. Use the universal gravitational constant $G \approx 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$.
- Taking the potential energy at infinity to be zero, what is the gravitational potential energy of the object on the Earth's surface?
- Taking the potential energy on the Earth's surface to be zero, what is the gravitational potential energy of the object on the surface?
U = -6.3 \times 10^7 J U = 0 J
The gravitational potential energy $U$ of a mass $m$ at a distance $r$ from the center of a larger mass $M$ is defined relative to a zero point.
[Q1] With the zero point at infinity ($U(\infty)=0$), the potential energy is:
$$U(r) = -\frac{GMm}{r}$$On the Earth's surface, $r = R_e$:
$$U(R_e) = -\frac{GMm}{R_e} = -\frac{(6.67 \times 10^{-11})(6.0 \times 10^{24})(1)}{6.4 \times 10^6} \text{ J} \approx -6.3 \times 10^7 \text{ J}$$[Q2] If the potential energy on the Earth's surface is defined as the zero point, then by definition, the potential energy of the object located on the surface is zero.