Physics equations/Impulse, momentum, and motion about a fixed axis

Impulse and momentum
Impulse is denoted by the symbols I (which is too easily confused with moment of inertia), or Imp, or J (we shall use the latter):


 * $$\vec J = F\Delta t\rightarrow\int_{t_1}^{t_2} \vec F (t) dt$$

Momentum (or linear momentum) is:
 * $$\vec{p} = m\vec v \rightarrow m_1 \vec{v_1} + m_2 \vec{v_2}\rightarrow\sum m_j \vec{v_j}$$

Total linear momentum is an extrinsic and conserved quantity, provided the net external force is zero. It can be shown that momentum obeys (d/dt)Σp=ΣFext. Kinetic energy and momentum are related by, K=½mv2=p2/(2m). Linear momentum is related to linear momentum by the impulse-momentum theorem:
 * $$\vec{J}=\vec{p_2} - \vec{p_1} = \Delta \vec{p}$$

From Newton's second law, force is related to momentum p by
 * $$\mathbf{F} = \frac{d\mathbf{p}}{dt}.$$. Therefore,
 * $$\begin{align}

\mathbf{J} &= \int_{t_1}^{t_2} \frac{d\mathbf{p}}{dt}\, dt \\ &= \int_{p_1}^{p_2} d\mathbf{p} \\ &= \mathbf{p_2} - \mathbf{p_1} = \Delta \mathbf{p}, \end{align}$$

where Δp is the change in linear momentum from time t1 to t2. This is often called the impulse-momentum theorem.

Torque
Torque, moment or moment of force is also called moment. The symbol for torque is typically τ, the Greek letter tau. When it is called moment, it is commonly denoted M. The for torque is the  (N·m). It would be inadvisable to call this a Joule, even though a Joule is also a (N·m).

More generally, the torque on a particle (which has the position r in some reference frame) can be defined as the cross product:
 * $$\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F},$$

where r is the particle's position vector relative to the fulcrum, and F is the force acting on the particle. The magnitude τ of the torque is given by
 * $$\tau = rF\sin\theta,\!$$

where r is the distance from the axis of rotation to the particle, F is the magnitude of the force applied, and θ is the angle between the position and force vectors. Alternatively,
 * $$\tau = rF_{\perp}=r_{\perp}F,$$

where F⊥ is the amount of force directed perpendicularly to the position of the particle and r⊥ is called the lever arm.

Rotational motion about a fixed axis


Angular displacement may be measured in radians or degrees. If using radians, it provides a very simple relationship between distance traveled around the circle and the distance r from the center.

A particle moves in a circle of radius $$r$$. Having moved an arc length $$s$$, its angular position is $$\theta$$ relative to its original position, where $$\theta=\frac{s}{r}$$.

In mathematics and physics it is usual to use the natural unit radians rather than degrees or revolutions. Units are converted as follows:


 * $$1 \mathrm{\ rev} = 360^{\circ} = 2\pi \mathrm{\ rad}$$

An angular displacement is described as


 * $$ \Delta \theta = \theta_{2} - \theta_{1}, \!$$

Angular speed and angular velocity
Angular velocity is the change in angular displacement per unit time. The symbol for angular velocity is $$\omega$$ and the units are typically rad s−1. Angular speed is the magnitude of angular velocity.


 * $$\overline{\omega} = \frac{\Delta \theta}{\Delta t} = \frac{\theta_2 - \theta_1}{t_2 - t_1}.$$

The instantaneous angular velocity is related to particles speed by


 * $$\omega(t) = \frac{d\theta}{dt}=\frac{v}{r},$$

where $$v$$ is the transitional speed of the particle. A changing angular velocity indicates the presence of an angular acceleration in rigid body, typically measured in rad s−2. The average angular acceleration $$\overline{\alpha}$$ over a time interval Δt is given by
 * $$\overline{\alpha} = \frac{\Delta \omega}{\Delta t} = \frac{\omega_2 - \omega_1}{t_2 - t_1}.$$

The instantaneous acceleration α(t) is given by
 * $$ \alpha(t) = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2} =\frac{a}{r}.$$

Kinematic equations of motion
When the angular acceleration is constant, the five quantities angular displacement $$\theta$$, initial angular velocity $$\omega_i$$, final angular velocity $$\omega_f$$, angular acceleration $$\alpha$$, and time $$t$$ can be related by four equations of kinematics:


 * $$\omega_f = \omega_i + \alpha t\;\!$$
 * $$\theta = \omega_i t + \begin{matrix}\frac{1}{2}\end{matrix} \alpha t^2 $$
 * $$\omega_f^2 = \omega_i^2 + 2 \alpha\theta$$
 * $$\theta = \tfrac{1}{2} \left(\omega_f + \omega_i\right) t$$

Kinetic energy
The kinetic energy of a rigid system of particles moving in the plane is given by


 * $$K = \sum_{i=1}^N \frac12\,m_i \mathbf{v}_i\cdot\mathbf{v}_i = \sum_{i=1}^N \frac12\,m_i (\omega r_i)^2 = \frac12\, \omega^2 \sum_{i=1}^N m_i r_i^2.$$

Thus, $$K = I \omega^2$$ where $$I = \sum_{i=1}^N m_i r_i^2.$$ is called the moment of inertia.

The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, with the summation replaced by the integral,
 * $$ I= \int_V \rho(\mathbf{r})\,\mathbf{r}^2 \, dV \rightarrow

\int_S \sigma(\mathbf{r})\,\mathbf{r}^2 \, dS\rightarrow \int_\ell \lambda(x)x^2\, d\ell $$

Here r is the distance to the axis and ρ=ρ(r) is the mass density. As shown above, this can be converted into a line, surface, or volume integral for a substance with a surface mass density ρ(x,y) or line mass density λ(x)

Torque, angular momentum, and work
The rotational equivalent of Newton's ,F = ma, linear momentum as, p = mv, and work as W = FΔx → ʃFdx, is $$\tau = I\alpha$$ , $$L = I\omega$$, and $$W = \tau\Delta\theta$$, respectively.Here, L is angular momentum, which does not have the same units as linear momentum. But work, W, is measured in the same units (Joules).