Systems of Particles & Linear Momentum

Linear Momentum

$\vec{p} = m\vec{v}$

Newton's Second Law (Momentum Form)

$\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$

For constant mass: $\vec{F} = m\vec{a}$.

Impulse (Calculus Form)

$\vec{J} = \int_{t_i}^{t_f} \vec{F}(t)\,dt = \Delta\vec{p}$

Center of Mass

Discrete system: $x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}$

Continuous body: $x_{\text{cm}} = \frac{1}{M}\int x\,dm$

For a uniform rod of length $L$: $x_{\text{cm}} = L/2$ (from one end).

Motion of the Center of Mass

$\vec{F}_{\text{ext}} = M\vec{a}_{\text{cm}}$ $\vec{p}_{\text{total}} = M\vec{v}_{\text{cm}}$

The center of mass moves as if all mass were concentrated there and all external forces acted on it.

Collisions

• Elastic: conserve both momentum and KE.
• Inelastic: conserve momentum only.

2D collisions: apply momentum conservation in both $x$ and $y$ components independently.

Rocket Propulsion (Variable Mass)

Thrust: $F_{\text{thrust}} = v_{\text{exhaust}} \frac{dm}{dt}$

$v(t) = v_0 + v_{\text{exhaust}} \ln\frac{m_0}{m(t)}$

Problem: A time-varying force $F(t) = 6t^2$ N acts on a $3\ \text{kg}$ mass from $t = 0$ to $t = 2\ \text{s}$. If the object starts from rest, find the final velocity.

Solution:

$J = \int_0^2 6t^2\,dt = 2t^3\Big|_0^2 = 16\ \text{N·s}$

$\Delta p = J = mv_f - 0$ $v_f = \frac{16}{3} \approx 5.33\ \text{m/s}$

• $\vec{F} = d\vec{p}/dt$ is the most general form of Newton's second law.
• Impulse: $J = \int F\,dt = \Delta p$.
• Center of mass: $x_{\text{cm}} = \frac{1}{M}\int x\,dm$ for continuous bodies.
• The center of mass of a system responds to external forces only.