Chapter 8: Problem 1
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
Step by step solution
Define the Center of Mass and Relative Position
Express Individual Positions in terms of CM and Relative Position
Derive Expressions for Velocities
Calculate Kinetic Energy of Each Particle
Simplify the Total Kinetic Energy Expression
Combine and Recognize the Reduced Mass Term
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Relative Position
- Helps in understanding the internal configuration of a system of particles.
- Provides the basis for defining other quantities like relative velocity and relative acceleration.
Reduced Mass
- 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.
Kinetic Energy
- 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.