Question

\({ }^{\dagger}\) Two particles \(P_{1}\) and \(P_{2}\) have respective masses \(m_{1}\) and \(m_{2}\) and are attracted to each other by a force with time- independent potential \(U(\boldsymbol{r})\), where \(\boldsymbol{r}\) is the vector from \(P_{1}\) to \(P_{2}\). (1) Show that the motion of the centre of mass is the same as in the case \(U=0\). (2) Show that the motion of \(P_{2}\) relative to \(P_{1}\) is the same as in the case that \(P_{1}\) is fixed and \(P_{2}\) is attracted to \(P_{1}\) by a force with potential $$ V=\frac{m_{1}+m_{2}}{m_{1}} U . $$

Short answer

Question: Show that the motion of the center of mass is the same as in the case where U=0, and find the motion of P2 relative to P1. Answer: The motion of the center of mass is the same as in the case where U=0 because the total force acting on the system is zero, and therefore, the total momentum of the system is conserved. The motion of P2 relative to P1 is the same as the case when P1 is fixed, and P2 is attracted to P1 by a force with potential V, where V = (m1 + m2) / m1 * U.

Step-by-step solution

Part 1: Motion of the center of mass.

To show that the motion of the center of mass is the same as in the case where U=0, we need to calculate the total momentum and then show that it is conserved. The total momentum of the system can be written as the sum of the momenta of the two particles: $$ \boldsymbol{P} = m_{1} \boldsymbol{v}_{1} + m_{2} \boldsymbol{v}_{2} $$ Differentiating the potential energy U(r) with respect to time t, and using the chain rule, we get: $$ \frac{dU}{dt} = \frac{\partial U}{\partial \boldsymbol{r}} \cdot \frac{d\boldsymbol{r}}{dt} = \boldsymbol{F}_{12} \cdot \frac{d\boldsymbol{r}}{dt} $$ where \(\boldsymbol{F}_{12}\) is the force acting on particle 1 due to particle 2, and \(\frac{d\boldsymbol{r}}{dt}\) is the relative velocity between the two particles. According to Newton's third law, the force acting on particle 2 due to particle 1 is equal and opposite to the force acting on particle 1 due to particle 2: \(\boldsymbol{F}_{21} = -\boldsymbol{F}_{12}\). Thus, the total force acting on the system is given by \(\boldsymbol{F}_{12} + \boldsymbol{F}_{21} = 0\). Since the total force acting on the system is zero, the total momentum of the system is conserved: $$ \frac{d\boldsymbol{P}}{dt} = m_{1} \frac{d\boldsymbol{v}_{1}}{dt} + m_{2} \frac{d\boldsymbol{v}_{2}}{dt} = m_{1} \boldsymbol{F}_{12} + m_{2} \boldsymbol{F}_{21} = 0 $$ This conservation of momentum is identical to the case where U=0. Therefore, the motion of the center of mass is the same as the case where U=0.

Part 2: Motion of P2 relative to P1.

To find the motion of P2 relative to P1, let's first introduce a new variable \(\boldsymbol{R}\) to represent the position of the center of mass: $$ \boldsymbol{R} = \frac{m_{1} \boldsymbol{r}_{1} + m_{2} \boldsymbol{r}_{2}}{m_{1} + m_{2}} $$ Differentiating the position of the center of mass with respect to time the, we get the velocity of the center of mass: $$ \frac{d\boldsymbol{R}}{dt} = \frac{m_{1} \frac{d\boldsymbol{r}_{1}}{dt} + m_{2} \frac{d\boldsymbol{r}_{2}}{dt}}{m_{1} + m_{2}} = \frac{\boldsymbol{P}}{m_{1} + m_{2}} $$ Now, we introduce a new variable \(\boldsymbol{u}\) to represent the position of P2 relative to P1: $$ \boldsymbol{u} = \boldsymbol{r}_{2} - \boldsymbol{r}_{1} $$ Taking the derivative of \(\boldsymbol{u}\) with respect to time, we get the relative velocity: $$ \frac{d\boldsymbol{u}}{dt} = \frac{d\boldsymbol{r}_{2}}{dt} - \frac{d\boldsymbol{r}_{1}}{dt} $$ Let's assume the potential V depends only on the relative position \(\boldsymbol{u}\). $$ \frac{dV}{dt} = \frac{\partial V}{\partial \boldsymbol{u}} \cdot \frac{d\boldsymbol{u}}{dt} = \left( \frac{m_{1} + m_{2}}{m_{1}} \right) \boldsymbol{F}_{12} \cdot \frac{d\boldsymbol{u}}{dt} $$ As the force acting on particle 2 due to particle 1 is equal and opposite to the force acting on particle 1 due to particle 2, we can write the force acting on particle 2 due to particle 1 as: $$ \boldsymbol{F}_{21} = -\boldsymbol{F}_{12} = \frac{\partial V}{\partial \boldsymbol{u}} $$ Hence, we have found the potential V that satisfies the condition: $$ V = \frac{m_{1} + m_{2}}{m_{1}} U $$ This shows that the motion of P2 relative to P1 is the same as the case when P1 is fixed, and P2 is attracted to P1 by a force with potential V.

Key concepts

Conservation of Momentum

The concept of Conservation of Momentum is a fundamental principle in physics that tells us that the total momentum of an isolated system remains constant over time, provided no external forces act on it. In the context of two particles, as given in the original exercise, the total momentum is the vector sum of the individual momenta of the particles, expressed as:
\[\boldsymbol{P} = m_{1} \boldsymbol{v}_{1} + m_{2} \boldsymbol{v}_{2}\]
When analyzing the motion of the center of mass, we realize that because Newton's third law ensures the internal forces between these two particles are equal and opposite, their effects cancel out each other, leading to zero net external force. Thus, the total momentum \(\boldsymbol{P}\) is conserved:

• The derivative \(\frac{d\boldsymbol{P}}{dt}\) remains zero when forces between the particles cancel out.
• This means \(\boldsymbol{P}\) is constant, similar to when the potential \(U = 0\).

Understanding this helps explain why the center of mass moves in the same way regardless of whether or not there's potential energy, as long as those forces remain internal to the system.

Newton's Third Law

Newton's Third Law, often summarized as "for every action, there is an equal and opposite reaction," is integral in understanding interactions between objects. In the scenario with two particles, this law implies that the force exerted by \(P_1\) on \(P_2\) through their mutual potential \(U(\boldsymbol{r})\) is equal in magnitude and opposite in direction to the force exerted by \(P_2\) on \(P_1\):
\[\boldsymbol{F}_{21} = -\boldsymbol{F}_{12}\]
This relationship means:

• The net force acting on the system of two particles is zero.
• Such forces do not alter the system's overall motion but only affect how the particles interact.

This principle is paramount in demonstrating the overall conservation of momentum since these internal forces manifest as activity between the particles but do not contribute to external movement. Thus, their collective center of mass behaves as if only external forces would change its motion.

Relative Motion Analysis

Relative motion analysis is a powerful tool in physics for understanding how the position and velocity of one object appear when viewed from another object's reference frame. In our exercise:

• The position of \(P_2\) relative to \(P_1\) is given as \(\boldsymbol{u} = \boldsymbol{r}_{2} - \boldsymbol{r}_{1}\).
• This means the relative velocity can be expressed as \(\frac{d\boldsymbol{u}}{dt} = \frac{d\boldsymbol{r}_{2}}{dt} - \frac{d\boldsymbol{r}_{1}}{dt}\).

By analyzing how \(P_2\) moves with respect to \(P_1\), we transform the two-body problem into a simpler one-body problem, treating \(P_1\) as a fixed point and analyzing \(P_2\)'s behavior under the adjusted potential \(V = \frac{m_{1} + m_{2}}{m_{1}} U\). This technique simplify calculations regarding forces and potentials, providing a clear understanding of interactions in relative motion.

Potential Energy

Potential energy is energy that is stored within a system due to its position or arrangement. In this exercise, the potential energy function \(U(\boldsymbol{r})\) is dependent on the separation \(\boldsymbol{r}\) between two particles. Analysis of potential energy allows us to:

• Understand the force between the particles as a gradient of the potential energy, \(\boldsymbol{F} = -abla U(\boldsymbol{r})\).
• Examine how a system's configuration and energy states change as the particle positions vary.

By introducing a new potential \(V\), altered by the masses \(m_1\) and \(m_2\), we describe how other effects might influence motion if we treat one particle (\(P_1\)) as fixed. This alternate potential aptly models how additional mass or constraints would affect gravitational-like attractive forces, showing the intricate balance of mass and potential within these systems.