Circular and Rotational Motion Practice Flashcards

Vocabulary-style flashcards based on lecture notes covering Kinematics, Dynamics, Rotational Motion, and Rolling Motion formulas and concepts.

Angular Velocity ($\omega$)

The rate of change of angular displacement, given by $\omega = \frac{d\theta}{dt} = 2\pi n = \frac{2\pi}{T}$, where $n$ is frequency and $T$ is the time period.

Relation between Linear and Angular Velocity

$v = r\omega$ (Vector form: $\mathbf{v} = \mathbf{\omega} \times \mathbf{r}$).

Centripetal Acceleration ($a_c$)

The acceleration directed toward the center of a circular path, expressed as $a_c = \frac{v^2}{r} = r\omega^2 = v\omega$.

Centripetal Force ($F_c$)

$F_c = \frac{mv^2}{r} = mr\omega^2 = mv\omega$ (Directed towards the center).

Max Safe Speed on Unbanked Road ($v_{max}$)

$v_{max} = \sqrt{\mu_s rg}$, where $\mu_s$ is the coefficient of static friction.

Angle of Banking ($\theta$)

$\theta = \tan^{-1}\left( \frac{v^2}{rg} \right)$.

Optimum Speed ($v_0$)

The ideal speed for a banked road to minimize wear on tires, given by $v_0 = \sqrt{rg \tan(\theta)}$.

Max Safe Speed on Banked Road (with friction)

$v_{max} = \sqrt{rg \left( \frac{\mu_s + \tan(\theta)}{1 - \mu_s \tan(\theta)} \right)}$.

Min Safe Speed on Banked Road (with friction)

$v_{min} = \sqrt{rg \left( \frac{\tan(\theta) - \mu_s}{1 + \mu_s \tan(\theta)} \right)}$.

Conical Pendulum Time Period ($T$)

$T = 2\pi \sqrt{\frac{L \cos(\theta)}{g}} = 2\pi \sqrt{\frac{h}{g}}$, where $L$ is the length and $h$ is the vertical height.

VCM Top Position (Highest)

Minimum velocity $v_{min} = \sqrt{rg}$ and Tension $T_{top} \geq 0$.

VCM Bottom Position (Lowest)

Minimum velocity $v_{min} = \sqrt{5rg}$ and Tension $T_{bottom} = 6mg$.

VCM Midway Position (Horizontal)

Minimum velocity $v_{min} = \sqrt{3rg}$ and Tension $T_{mid} = 3mg$.

Moment of Inertia (M.I.) ($I$)

The sum of the products of the mass of each particle and the square of its distance from the axis of rotation: $I = \sum m_i r_i^2 = \int r^2 dm$.

Radius of Gyration ($k$)

$k = \sqrt{\frac{I}{M}}$; the distance from the axis where the entire mass can be assumed to be concentrated.

Parallel Axes Theorem

$I_0 = I_c + Mh^2$, where $I_c$ is the moment of inertia through the center of mass and $h$ is the distance between axes.

Perpendicular Axes Theorem

$I_z = I_x + I_y$ (Applicable for planar bodies only).

M.I. of a Ring ($I_c$)

$MR^2$ (Axis through center, perpendicular to plane).

M.I. of a Disc ($I_c$)

$MR^2$.

M.I. of a Solid Cylinder ($I_c$)

$MR^2$ (Same as disc).

M.I. of a Hollow Cylinder ($I_c$)

$MR^2$ (Same as ring).

M.I. of a Solid Sphere ($I_c$)

$MR^2$.

M.I. of a Hollow Sphere / Shell ($I_c$)

$MR^2$.

M.I. of a Thin Rod ($I_c$)

$\frac{1}{12} ML^2$ (Axis through center, perpendicular to length).

Torque ($\tau$)

$\tau = I\alpha$ (Vector form: $\mathbf{\tau} = \mathbf{r} \times \mathbf{F}$).

Angular Momentum ($L$)

$L = I\omega$ (Vector form: $\mathbf{L} = \mathbf{r} \times \mathbf{p}$).

Law of Conservation of Angular Momentum

If net external torque $\tau_{ext} = 0$, then $L = \text{constant}$, or $I_1\omega_1 = I_2\omega_2$.

Rotational Kinetic Energy ($E_r$)

$E_r = \frac{1}{2} I \omega^2 = \frac{L^2}{2I}$.

Work Done in Rotation ($W$)

$W = \tau \theta$.

Total Kinetic Energy ($E_T$) in Rolling

The sum of translational and rotational kinetic energy: $E_T = E_t + E_r = \frac{1}{2} mv^2 + \frac{1}{2} I\omega^2$.

Acceleration Area of a body rolling down an Inclined Plane

$a = \frac{g \sin(\theta)}{1 + \frac{k^2}{R^2}}$, where $\theta$ is the angle of the incline.

Velocity at bottom of an Inclined Plane

$v = \sqrt{\frac{2gh}{1 + \frac{k^2}{R^2}}}$.