Translational and Rotational Motion

Flashcards covering definitions, rotational analogs, kinematics, dynamics, and equilibrium conditions for translational and rotational motion.

Translational Motion

Linear motion in one, two, or three dimensions characterized by the movement of an object from one point to another through space.

Rotational Motion

The motion of an object turning about an axis, where each particle of the object moves along a circular path.

Rotational Analog

A quantity in rotational motion that corresponds to a specific quantity in linear motion; all linear quantities have these counterparts.

Mechanics

The branch of physics that involves the study of motion.

Kinematics

The mathematical description of motion.

Dynamics

The study of the causes that produce motion.

Angular Displacement ($\theta$)

The change in angular position represented as $\Delta \theta = \theta_f - \theta_o$, typically measured in $radians$.

$$1 \text{ revolution}$$

A unit of rotation equivalent to $360^o$, $2\pi \text{ radians}$, or approximately $6.28 \text{ radians}$.

Linear Distance ($d$)

Also known as arc length, it is the distance in meters ($m$) along an arc subtended by an angle $\theta$ (in radians) in a circle of radius $r$, given by $d = \theta r$.

Angular Velocity ($\omega$)

The rate at which angular displacement changes with time, where average angular velocity is $\bar{\omega} = \frac{\Delta \theta}{\Delta t}$ with the unit $rad/s$.

Tangential Velocity ($v$)

The linear velocity of a point on a rotating rigid body, expressed as $v = \omega r$.

Angular Acceleration ($\alpha$)

The time rate of change of angular velocity, calculated as $\alpha = \frac{\omega_f - \omega_o}{t}$ and measured in $rad/s^2$.

Tangential Acceleration ($a$)

The linear acceleration of a point on a rotating object, calculated by $a = \alpha r$.

Radial Acceleration ($a_{rad}$)

Also known as centripetal acceleration, defined by the formula $a_{rad} = \frac{v^2}{r}$.

Torque ($\tau$)

The rotational analog of force ($F$), representing the ability of a force to rotate an object; it is the product of force and the moment arm: $\tau = d_{\perp}F$.

Moment Arm

Also called the lever arm ($d_{\perp}$), it is the perpendicular ($90^o$) distance from the axis of rotation to the line of action of the force.

Net Torque ($\tau_{net}$)

The sum of all individual torques acting on an object, where clockwise rotation is considered negative ( $-$ ) and counterclockwise rotation is positive ( $+$ ).

Moment of Inertia ($I$)

The rotational analog of mass ($m$); it is the property of a rotating object to resist a change in its state of rotation, measured in $kg \cdot m^2$.

Angular Momentum ($L$)

The quantity of rotational motion an object possesses, defined as $L = I\omega$ and measured in $$kg \cdot m^2/s$ flavour.

Newton's Second Law for Rotation

States that an unbalanced torque produces an angular acceleration directly proportional to the torque and inversely proportional to the moment of inertia, expressed as $\tau = I\alpha$.

Translational Equilibrium

A condition occurring when the net force acting on an object is zero ($\Sigma F = 0$), resulting in straight-line motion at constant velocity.

Rotational Equilibrium

A condition occurring when the net torque acting on an object is zero ($\Sigma \tau = 0$), resulting in a constant angular speed with no angular acceleration.