Question

A system consists of three particles, each of mass $\mathbf{5}\mathbf{.}\mathbf{00}\mathit{g}$, located at the corners of an equilateral triangle with sides of $\mathbf{30}\mathbf{.}\mathbf{0}\mathit{c}\mathit{m}$. (a) Calculate the potential energy of the system. (b) Assume the particles are released simultaneously. Describe the subsequent motion of each. Will any collisions take place? Explain.

Short answer

(a) Total potential energy of the system will be$U_{Total}=-1.67\times {10}^{-17}\mathrm{J}$

(b) All particles will collide with each other at the center of the triangle.

Step-by-step solution

Conservation of energy

The conservation of energy says that’s the total energy in the initial condition of the object will be equal to the total energy in the final condition.

$\begin{array}{rcl}{\left(K+U\right)}_{initial}& =& {\left(K+U\right)}_{final}\\ K& =& kineticenergy\\ U& =& Potentialenergy\end{array}$

Given

$$\begin{gathered} \text{massm=5.00g} \\ \text{sidesofthetriangle=30.0cm} \end{gathered}$$

(a) Potential energy of the system

As described in the question the particles are located at the corners of the triangle and this triangle have equal three sides that is equilateral triangle.

The length of sides is 0.30m, so the gravitational potential energy of the system will be, if triangle is ABC,

$$\begin{gathered} U_{Total}=U_{AB}+U_{BC}+U_{CA}=3U_{AB} \\ {U}_{Total}=3\left(-\frac{G{m}_A{m}_B}{r_{AB}}\right) \\ {U}_{Total}=3\left(-\frac{6.67\times {10}^{-11}\text{N}-{\text{m}}^2{\text{/Kg×5.00×10}}^{-3}{\text{Kg×5.00×10}}^{-3}\text{Kg}}{0.300\text{m}}\right) \\ {U}_{Total}=-1.67\times {10}^{-14}\text{J} \end{gathered}$$

So the Total potential energy of the system will be$U_{Total}=-1.67\times {10}^{-14}\text{J}$.

(b) Motion of the particles

The net force of attraction felt by a particle will be at the midpoint of the other two points of the triangle. Each particle of the triangle is moving towards the center point of the triangle, so the all particles will collide with each other at the center of the triangle.