Work, Energy, Impulse, and Momentum

Work and Energy

Work Done by a Force

$W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

where θ is the angle between the force and displacement vectors.

Work done by a spring: W = -½k(x₂² - x₁²)

Kinetic Energy

$KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

Potential Energy

Gravitational: PE = mgh (h = height above reference datum)

Elastic (spring): PE = ½kx² (x = deformation from natural length)

Work-Energy Theorem

$\sum W = \Delta KE = KE_2 - KE_1$

Conservation of Energy

When only conservative forces (gravity, springs) do work:

$KE_1 + PE_1 = KE_2 + PE_2$

$\frac{1}{2}mv_1^2 + mgh_1 + \frac{1}{2}kx_1^2 = \frac{1}{2}mv_2^2 + mgh_2 + \frac{1}{2}kx_2^2$

Power

$P = \frac{dW}{dt} = \vec{F} \cdot \vec{v} = Fv\cos\theta$

For rotational motion: P = τω (torque × angular velocity)

Impulse and Momentum

Linear Impulse-Momentum Theorem

$\int_{t_1}^{t_2} \vec{F} \, dt = m\vec{v}_2 - m\vec{v}_1$

For constant force: F·Δt = m(v₂ - v₁)

Conservation of Linear Momentum

When no external impulse acts on the system: $m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$

Collisions

Coefficient of Restitution: $e = \frac{v_2' - v_1'}{v_1 - v_2} = \frac{\text{relative velocity of separation}}{\text{relative velocity of approach}}$

For perfectly plastic collisions: $m_1 v_1 + m_2 v_2 = (m_1 + m_2) v'$

Angular Impulse-Momentum

$\int_{t_1}^{t_2} M \, dt = I\omega_2 - I\omega_1$

For constant moment: M·Δt = I(ω₂ - ω₁)

Vibrations (Natural Frequency)

For an undamped spring-mass system:

Natural frequency: $\omega_n = \sqrt{\frac{k}{m}} \quad (\text{rad/s})$

Period: $T = \frac{2\pi}{\omega_n} = 2\pi\sqrt{\frac{m}{k}} \quad (\text{seconds})$

Frequency: $f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{k}{m}} \quad (\text{Hz})$

For a simple pendulum: $\omega_n = \sqrt{\frac{g}{L}} \quad T = 2\pi\sqrt{\frac{L}{g}}$