AP Physics Unit 5/6 Formulas

Arc length formula

S = r\theta

Linear velocity formula

\overrightarrow{v} = r\overrightarrow{\omega}

Linear (tangential) acceleration formula

\overrightarrow{a_c} = r\alpha

\theta (rad) to \theta (deg) formula

\theta (rad) = \frac{\pi}{180}\theta (deg) 

Points on spinning objects have the same (angular/linear) velocities but not the same (angular/linear) velocities

angular, linear

\omega_A = \omega_B = \omega_C

but

v_A \neq v_B \neq v_C

Formula for average angular velocity?

\overrightarrow{\omega_{avg}} = \frac{\theta_f - \theta_i}{t_f - t_i} 

Formula for average angular acceleration?

\alpha_{avg} = \frac{\omega_f - \omega_i}{t_f - t_i}

What are the four rotational kinematics equations?

\omega_f = \omega_i + \alpha t

\theta_f = \theta_i + \omega_it + \frac{1}{2}\alpha t²

\omega_f² = \omega_i² + (2)(\alpha)(\theta_f - \theta_i)

\theta_f = \theta_i + \frac{1}{2}(\omega_i + \omega_f)t

For rotational kinematics, is \alpha constant or not?

constant

Formula for total acceleration?

\overrightarrow{a} = \overrightarrow{a_T} + \overrightarrow{a_c} 

Formula for magnitude of total acceleration?

||\overrightarrow{a}|| = \sqrt{a_T² + a_c²}

Formula for torque?

\tau = rFsin(\phi) OR \alpha \int r²dm

What does \phi mean in rotation and torque?

Angle of force with the “moment arm”

What is the formula for net torque?

\sum \tau = I\alpha

What is the formula for d?

d = rsin(\phi)

What does d mean in rotation and torque?

the moment arm (perpendicular distance from the axis of rotation to a line drawn in the direction of force)

Sum of all torques on a system

\sum \tau = \tau_1 + \tau_2 + … + \tau_n = F_1d_1 + F_2d_2 + … + F_nd_n 

Formula for tangential force of a rotating object

\overrightarrow{F_t} = ma_t

(for pulley systems) formula for the sum of y-forces?

\sum F_y = ma_y = mg - T

Formula for I ?

I = \int r²dm

(in equilibrium) sum of x-directional forces

\sum \overrightarrow{F_x} = 0

(in equilibrium) sum of y-directional forces

\sum \overrightarrow{F_y} = 0

(in equilibrium) sum of torque forces

\sum \tau = 0

Equation for volume density?

\rho = \frac{M}{V}

Equation for surface density?

\sigma = \frac{M}{A} 

Equation for linear density?

\lambda = \frac{M}{L}

Equation for parallel axis theorem?

I_{tot} = I_{cm} + MD²

What does “D” represent in the parallel-axis theorem?

The distance the axis of rotation has moved compared to the center of mass

When is angular momentum conserved?

When there are no external torques

True/false: angular momentum is conserved even with an external force

True, so long as no torque is produced

Formula(s) for angular momentum?

\overrightarrow{L} = \overrightarrow{r} \times \overrightarrow{p}

\overrightarrow{L} = I\overrightarrow{\omega}

First is used for non-rigid systems

Second is used for rigid-body systems

Formula for torque using \overrightarrow{L}

\sum \tau = \frac{d\overrightarrow{L}}{dt}

Torque formula using linear monentum?

\sum \tau = \overrightarrow{r} \times \frac{d\overrightarrow{p}}{dt}

Magnitude of angular momentum

L = mvrsin(\phi)

In what direction is \overrightarrow{L} in relative to \overrightarrow{r} and \overrightarrow{p}?

Perpendicular to the plane of r and p

True/false: A system can be isolated in energy, linear momentum, and angular momentum

true

True/false: A system can be isolated in one aspect but not another

true

Formula for conservation of angular momentum?

I\omega_1 = I\omega_2

Angular analog for centripetal force?

F_c = m\omega²r

Angular analog of power?

P = \tau \omega

Average work formula?

P = \frac{W}{T}

Rotational inertia for a hoop (hollow cylinder)

I = MR²

Rotational inertia for a cylinder with a hole in it?

I = \frac{1}{2}(R_1² + R_2²)

Rotational inertia for a sphere with uniform mass?

I = \frac{1}{2}MR²

Rotational inertia for a thin rod rotating from its center of mass?

I = \frac{1}{12}ML²

Rotational inertia for a sphere with uniform mass?

I = \frac{2}{5}MR²

Rotational inertia of a thin spherical shell?

I = \frac{2}{3}MR²

Rotational inertia for a plane spinning through its center of mass?

I = \frac{1}{12}M(a²+b²)

Rotational inertia for a rod spinning about its end?

I = \frac{1}{3}MR²