Newton’s laws, connected particles and impulse

What you’ll learn

• Apply force equations to individual particles and connected systems.
• Use momentum change to calculate impulse.

Understand the idea

Newton's second law uses the resultant force, so draw a force diagram and resolve in a chosen direction before writing $F=ma$. On an inclined plane, resolve weight parallel and perpendicular to the plane. An ideal light inextensible string over a smooth pulley has equal tension throughout and relates the particles' acceleration magnitudes.

For connected particles, separate equations identify internal forces such as tension; an equation for the whole system can eliminate them. Momentum is mass times velocity. During a short interaction, the resultant impulse equals the change in momentum, with directions retained in the signs.

Key formulae

$$\sum F=ma$$

$$\mathbf p=m\mathbf v$$

$$\mathbf I=\Delta\mathbf p=m(\mathbf v-\mathbf u)$$

$$I=F\Delta t\quad\text{(constant force)}$$

Worked example

A $3\,\mathrm{kg}$ particle on a smooth horizontal surface is pulled by $15\,\mathrm N$. Find its acceleration.

1. Weight and normal reaction cancel vertically.
2. The horizontal resultant is $15\,\mathrm N$.
3. Apply $15=3a$ and divide by $3$.

Answer: $5\,\mathrm{m\,s^{-2}}$ along the pull.