Centre of Mass and the Motion of a System: Quick Revision

A brief note on Centre of Mass and the Motion of a System. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Centre of mass: $x_{cm} = \dfrac{\sum m_ix_i}{\sum m_i}$, and the same for $y$, $z$. Nearer the heavier parts; can lie outside the material.
• Only external forces move it: $M\vec{a}_{cm} = \vec{F}_{\text{ext}}$. Internal forces, however violent, cannot.
• Momentum of a system: $\vec{P} = M\vec{v}_{cm}$; conserved with no external force.
• Rigid-body rotation: all particles share $\omega$ and $\alpha$; a particle at $r$ has $v = \omega r$.
• Rotation equations (steady $\alpha$): $\omega = \omega_0 + \alpha t$, $\theta = \omega_0t + \tfrac12\alpha t^2$, $\omega^2 = \omega_0^2 + 2\alpha\theta$.

The facts at a glance

• Two masses a distance $d$ apart: $x_{cm} = \dfrac{m_2d}{m_1 + m_2}$ from $m_1$
• A hole in a body: treat it as a negative mass
• Walking $L$ along a boat: boat moves $\dfrac{mL}{m + M}$
• Shell bursts in flight: centre of mass keeps the old parabola
• Rpm to rad/s: multiply by $\dfrac{2\pi\ \text{rad}}{60\ \text{s}}$

A worked example

A T-shaped sign is made of a $6.0\ \text{kg}$ crossbar, whose centre is $1.2\ \text{m}$ above the ground, on a $4.0\ \text{kg}$ upright post, whose centre is $0.50\ \text{m}$ up.

1. Set up. Both centres lie on one vertical line, so only $y$ matters.
2. Average. $\sum m_iy_i = 6.0\ \text{kg} \times 1.2\ \text{m} + 4.0\ \text{kg} \times 0.50\ \text{m} = 9.2\ \text{kg m}$, so $y_{cm} = \dfrac{9.2\ \text{kg m}}{10\ \text{kg}} = 0.92\ \text{m}$.
3. Sense check. It is nearer the heavier crossbar, as expected.

Practice with new numbers

A program to try

The program finds the centre of mass of the T-shaped sign.

Mistakes to avoid

• Averaging positions without the masses. Each position is weighted by its mass.
• Dividing by the wrong total. With a hole, divide by the mass that is left.
• Internal forces moving the centre of mass. They cannot.
• Turns in the rotation equations. Use radians.
• Different $\omega$ for different points. A rigid body has one $\omega$; it is $v$ that changes with $r$.

Check yourself

1. Masses of $1.0\ \text{kg}$ and $3.0\ \text{kg}$ are $2.0\ \text{m}$ apart. How far is the centre of mass from the $1.0\ \text{kg}$ mass?

2. A $50\ \text{kg}$ person walks $3.0\ \text{m}$ along a $200\ \text{kg}$ raft. How far does the raft move?

3. A wheel turning at $8.0\ \text{rad/s}$ speeds up steadily to $20\ \text{rad/s}$ in $4.0\ \text{s}$. Through what angle does it turn?

4. Why does a car's speedometer, which counts wheel turns, read high if the tyres are replaced by smaller ones?

5. A rocket in deep space fires its engine. Does the centre of mass of 'rocket plus exhaust gas' move?