Readings

• Fundamentals of Physics 1 AU/NZ: 9-1, 9-2
• Edition 11: 9-1, 9-2
• Edition 9: 9-1, 9-2, 9-3

Dynamics – Pre-Lecture 11

• System of Particles and Centre of Mass

Key Concepts

Work and Acceleration

• Work: A process of energy transfer to or from an object via the mechanism of an applied force.
• Acceleration: A process of change in the motion of an object via the mechanism of an applied force.

Formulas

• Acceleration:
  $extbf{a} = \frac{ extbf{F}}{m}$
• Work:
  $W = \frac{1}{2}(F ext{ displacement}) = extstyle ext{integral from } x_1 ext{ to } x_2 ext{ of } extbf{F} \bullet d extbf{x}$

Force and Motion

• Force: Transfer of Energy and Motion
• Discussion of point mass vs particle system:
  • Requires generalization of Newton’s Laws of Motion.

Particle System Dynamics

Definition

• A particle system can include:
  • Rigid bodies: Objects with fixed structures
  • Non-rigid bodies: Objects with structures that can change
  • Collections of separate particles: Can interact either through forces at a distance or via direct collisions

General Considerations

• Particle systems incorporate all these cases to facilitate the dynamics study.

Centre of Mass

Definition

• A centre of mass of a particle system is a special position that behaves as if all mass were concentrated there. All external forces on the particle system affect motion as if acting on the centre of mass.

Two Point Masses

Position Calculation

• Given two point masses $m_1$ and $m_2$ located at positions $x_1$ and $x_2$:
  • Average mass:
    $m_{ave} = m_1 + m_2$
• Average position (coordinate):
  $x_{ave} = \frac{x_1 + x_2}{2}$

Centre of Mass Formula

• Centre of mass position for two point masses:
  $x_{CM} = \frac{m_1 imes x_1 + m_2 imes x_2}{m_1 + m_2}$

Special Cases

• When $m_1 = m_2$, then
  $x_{CM} = x_{ave}$
• When one mass is zero ($m_2=0$):
  $x_{CM} = x_1$
• When the first mass is zero ($m_1=0$):
  $x_{CM} = x_2$

Center of Mass for N Point Masses

General Formula

• For particle systems comprising $N$ point masses:
  $x_{CM} = \frac{ ext{(total mass) }}{M} imes ext{(position contribution summation)}$
• Specifically:
  $x_{CM} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + … + m_N x_N}{m_1 + m_2 + m_3 + … + m_N}$

Three-Dimensional Centre of Mass

• If distributed in three dimensions, the coordinates of the centre of mass are given by:
  • $x_{CM} = \frac{1}{M} imes extstyle ext{(sum of } m_i imes x_i$
  • $y_{CM} = \frac{1}{M} imes extstyle ext{(sum of } m_i imes y_i$
  • $z_{CM} = \frac{1}{M} imes extstyle ext{(sum of } m_i imes z_i$

Definitions

• Where $M = extstyle ext{(sum of all masses)} = extstyle ext{Total mass of system}.$

Centre of Mass: Solid Bodies

Approach for Solid Bodies

• The centre of mass of solid bodies can be calculated similarly through integrals:
  • $x_{CM} = \frac{1}{M} imes extstyle ext{(integral over density)}$
• Constitutive equations involve:
  • 
    ho = rac{ ext{density}}{ ext{volume}} with $ ext{density}$ and volume leading to:
  • x_{CM} = rac{1}{M} imes extstyle ext{(integral of } x
    ho ext{dV)}
• This integral process applies to $y_{CM}$ and $z_{CM}$ respectively to find all coordinates efficiently.

Motion of Centre of Mass

• The inertia and motion can be treated using dynamics techniques previously developed in particle dynamics for the entire system.

Laws of Motion

• The motion of centre of mass follows:
  • $extbf{r}_{CM} = \frac{1}{M} imes extstyle (m_1 extbf{r}_1 + m_2 extbf{r}_2 + … + m_N extbf{r}_N)$
• This form allows us to visualize how all external forces determine the motion of the system.

Acceleration and Forces

• The acceleration of the centre of mass can be expressed as:
  $\frac{d extbf{r}<em>{CM}}{dt} = extbf{v}</em>{CM}$
• Therefore:
  $extbf{a}<em>{CM} = \frac{d extbf{v}</em>{CM}}{dt} = \frac{1}{M}(m_1 extbf{a}_1 + m_2 extbf{a}_2 + … + m_N extbf{a}_N)$
• Equations define the relationships between total external forces and acceleration of the system, leading to:
  $extbf{F}<em>{CM} = M extbf{a}</em>{CM} = extstyle ext{(sum of all external forces)}$

Newton's Third Law Within the System

• Discussion on internal interactions:
  • Internal forces cancel due to Newton's third law, producing a net effect of external forces on the centre of mass.

Implications of the Laws of Motion for Centre of Mass

• If the total external force $F_{external} = 0 $, then:
  • The centre of mass remains constant in velocity:
    $extbf{v}_{CM} = ext{constant or } = 0$

Conclusion

• Therefore using Newton's laws, one can predict the behaviour of the centre of mass just like a single point mass acting under all resultant external forces. This provides a straightforward method for analyzing multi-body systems using fundamental laws derived from single-body systems.