Question

(a) What is the gravitational potential energy of the two-particle system in Problem 3? If you triple the separation between theparticles, how much work is done (b) by the gravitational force between the particles and (c) by you?

Short answer

1. The gravitational potential energy is ${\text{- 4.4×10}}^{\text{- 11}}\text{J}$.
2. The work is done if the separation distance is tripled which is$\text{- 2.9}\times {\text{10}}^{\text{- 11}}\text{J}$.
3. Work done by you is$\text{2.9}\times {\text{10}}^{\text{- 11}}\text{J}$.

Step-by-step solution

Step 1: Given

Mass of particle,${\text{m}}_{\text{1}}=5.2\text{kg}$

Mass of particle,${\text{m}}_{\text{2}}=2.4\text{kg}$

The gravitational force of attraction is ${\text{2.3×10}}^{\text{- 12}} \text{N}$${\text{2.3×10}}^{\text{- 12}} \text{N}$

Determining the concept

Using the formula for gravitational force, find the distance between the particles. Use this formula to find the potential energy in terms of masses and the separation between the masses. Work done is equal to change in potential energy.

Formulae are as follows:

$$\begin{gathered} F=\frac{GMm}{r^2} \\ U=-\frac{GMm}{r} \end{gathered}$$

where, M, and m are masses, r is the radius, G is the gravitational constant, F is the gravitational force and U is potential energy.

(a) Determining the gravitational potential energy

Gravitational force is written as,

$$\begin{gathered} F=\frac{GMm}{r^2} \\ 2.3\times {10}^{-12}N=6.67\times {10}^{-11}N·m^2·k{g}^{-2}\times 5.2kg\times \frac{2.4kg}{r^2} \\ r=\sqrt{\frac{6.67\times {10}^{-11}N·m^2·k{g}^{-2}\times 5.2kg\times 2.4kg}{2.3\times {10}^{-12}N}} \\ r=19m \end{gathered}$$

Gravitational potential energy is written as,

$\begin{array}{rcl}U& =& -\frac{GMm}{r}\\ & =& -6.67\times {10}^{-11}N·m^2·k{g}^{-2}\times 5.2kg\times \frac{2.4kg}{{\left(19m\right)}^2}\\ & =& -4.4\times {10}^{-11}J\\ & & \end{array}$

Hence, the gravitational potential energy is ${\text{- 4.4×10}}^{\text{- 11}}\text{J}$.

(b) Determining the work done if the separation distance is tripled

Work done means a change in potential energy. So,

$$\begin{gathered} \Delta U=-\frac{GMm}{3r}-\left(-\frac{GMm}{r}\right) \\ =-6.67\times {10}^{-11}N·m^2·k{g}^{-2}\times 5.2kg\times \frac{2.8kg}{3\times 19m} \\ +6.67\times {10}^{-11}N·m^2·k{g}^{-2}\times 5.2kg\times \frac{2.8kg}{19m} \\ =3.41\times {10}^{-11}J \end{gathered}$$

Hence, the work done by the gravitational force is$\Delta U=-W=-2.9\times {10}^{-11}J$ .

(c) Determining the work done by you

Work done by you is given as,

$W=\Delta U$

role="math" localid="1661184490527" $\text{W}=2.9\times {10}^{-11}J$

Hence, thework done by you is.$2.9\times {10}^{-11}J$

Using Newton’s law of gravitation, the work done by gravitational force as well as by the person can be found.