Question

Verify that the positions of two particles can be written in terms of the CM and relative positions as \(\mathbf{r}_{1}=\mathbf{R}+m_{2} \mathbf{r} /\mathbf{M}\) and \(\mathbf{r}_{2}=\mathbf{R}-m_{1} \mathbf{r} / M .\) Hence confirm that the total \(\mathrm{KE}\) of the two particles can be expressed as \(T=\frac{1}{2} \mathbf{M} \dot{\mathbf{R}}^{2}+\frac{1}{2} \mu \dot{\mathbf{r}}^{2},\) where \(\mu\) denotes the reduced mass \(\mu=\mathbf{m}_{1} \mathbf{m}_{2} / \mathbf{M}.\)

Short answer

The positions and kinetic energy relations are verified, showing that the total KE can be split into CM and relative parts.

Step-by-step solution

Define the Center of Mass and Relative Position

The center of mass (CM) position \( \mathbf{R} \) for two particles is given by the equation: \[ \mathbf{R} = \frac{m_1 \mathbf{r}_1 + m_2 \mathbf{r}_2}{M} \] where \( M = m_1 + m_2 \) is the total mass of the system. The relative position \( \mathbf{r} \) between the two particles is defined as: \[ \mathbf{r} = \mathbf{r}_1 - \mathbf{r}_2 \]

Express Individual Positions in terms of CM and Relative Position

We need to express \( \mathbf{r}_1 \) and \( \mathbf{r}_2 \) in terms of \( \mathbf{R} \) and \( \mathbf{r} \). From the definitions in Step 1 and solving for \( \mathbf{r}_1 \) and \( \mathbf{r}_2 \): \[ m_1 \mathbf{r}_1 = m_1 \mathbf{R} + m_1 m_2 \mathbf{r}/M \]\[ m_2 \mathbf{r}_2 = m_2 \mathbf{R} - m_1 m_2 \mathbf{r}/M \]From which,\[ \mathbf{r}_1 = \mathbf{R} + \frac{m_2 \mathbf{r}}{M} \]\[ \mathbf{r}_2 = \mathbf{R} - \frac{m_1 \mathbf{r}}{M} \]

Derive Expressions for Velocities

Differentiate the expressions for \( \mathbf{r}_1 \) and \( \mathbf{r}_2 \) with respect to time to get the velocities:\[ \dot{\mathbf{r}}_1 = \dot{\mathbf{R}} + \frac{m_2 \dot{\mathbf{r}}}{M} \]\[ \dot{\mathbf{r}}_2 = \dot{\mathbf{R}} - \frac{m_1 \dot{\mathbf{r}}}{M} \]

Calculate Kinetic Energy of Each Particle

Use the velocity expressions to calculate the kinetic energy (KE) of the particles:\[ KE_1 = \frac{1}{2} m_1 \dot{\mathbf{r}}_1^2 = \frac{1}{2} m_1 \left( \dot{\mathbf{R}} + \frac{m_2 \dot{\mathbf{r}}}{M} \right)^2 \]\[ KE_2 = \frac{1}{2} m_2 \dot{\mathbf{r}}_2^2 = \frac{1}{2} m_2 \left( \dot{\mathbf{R}} - \frac{m_1 \dot{\mathbf{r}}}{M} \right)^2 \]

Simplify the Total Kinetic Energy Expression

The total kinetic energy (T) is the sum of \( KE_1 \) and \( KE_2 \).Substitute the expression \( M = m_1 + m_2 \) and simplify:\[ T = KE_1 + KE_2 = \frac{1}{2} m_1 \left( \dot{\mathbf{R}}^2 + 2\dot{\mathbf{R}} \frac{m_2 \dot{\mathbf{r}}}{M} + \left( \frac{m_2 \dot{\mathbf{r}}}{M} \right)^2 \right) + \frac{1}{2} m_2 \left( \dot{\mathbf{R}}^2 - 2\dot{\mathbf{R}} \frac{m_1 \dot{\mathbf{r}}}{M} + \left( \frac{m_1 \dot{\mathbf{r}}}{M} \right)^2 \right) \]

Combine and Recognize the Reduced Mass Term

Combine like terms and factor expressions:\[ T = \frac{1}{2} M \dot{\mathbf{R}}^2 + \frac{1}{2} \left( \frac{m_1 m_2}{M} \right) \dot{\mathbf{r}}^2 \]Recognize that \( \mu = \frac{m_1 m_2}{M} \), the reduced mass, thus:\[ T = \frac{1}{2} M \dot{\mathbf{R}}^2 + \frac{1}{2} \mu \dot{\mathbf{r}}^2 \]

Key concepts

Relative Position

The relative position of two particles in a system is a fundamental concept in physics. Imagine two particles moving in space, each with their respective positions \(\mathbf{r}_1\) and \(\mathbf{r}_2\). The relative position \(\mathbf{r}\) is simply defined as the difference between these two positions: \[ \mathbf{r} = \mathbf{r}_1 - \mathbf{r}_2 \] This vector tells us how far one particle is from the other and in which direction. It's a way to look at the system from the perspective of one particle with respect to the other.

• Helps in understanding the internal configuration of a system of particles.
• Provides the basis for defining other quantities like relative velocity and relative acceleration.

By using the relative position, we can express each particle's location as a combination of the center of mass (an average position of the entire system) and their separation. Recognizing and using \(\mathbf{r}\) allows for the simplification of complex equations involving multiple particle systems.

Reduced Mass

Reduced mass is an innovative concept that simplifies the study of two-body problems, such as the motion of electrons around a nucleus. When dealing with interactions between two particles with masses \(m_1\) and \(m_2\), the actual system behaves as if it were a single particle with a mass called the reduced mass \(\mu\). It is defined as: \[ \mu = \frac{m_1 m_2}{M} \] where \(M\) is the total mass \(M = m_1 + m_2\). This concept is particularly useful because it allows us to transform a complicated two-body problem into a simpler single-body problem.

• The reduced mass appears in the equations of motion and energy conservation for two-body systems.
• It provides insights into how forces and energy are distributed between interacting objects.

By using the reduced mass, physicists and engineers can solve problems like orbital mechanics and molecular dynamics more efficiently, gaining clear insights into the system's behavior without getting bogged down by complicated interactions between the two masses.

Kinetic Energy

Kinetic energy (KE) is a measure of how much energy an object has due to its motion. In the case of two particles, each one contributes to the total kinetic energy of the system. For each particle, the kinetic energy is given by: \[ KE = \frac{1}{2} m v^2 \] where \(m\) is the mass of the particle and \(v\) its velocity. When considering a system of two particles, we derive an expression for the total kinetic energy \(T\) by considering the center of mass and relative velocities: \[ T = \frac{1}{2} M \dot{\mathbf{R}}^2 + \frac{1}{2} \mu \dot{\mathbf{r}}^2 \]

• The first term represents the kinetic energy due to the motion of the system's center of mass.
• The second term captures the energy contribution from the relative motion between the particles, involving the reduced mass.

This division into two parts makes it easier to analyze systems with multiple particles by separating external movement from internal dynamics. Understanding these components helps in studying dynamics in mechanics, chemistry, and even quantum physics.