Chapter 8: Rotational Motion

Practice flashcards covering rotational kinematics, moment of inertia, rotational kinetic energy, torque, and angular momentum based on Chapter 8 of PHY 121.

Radian

The ratio between the length of an arc and its radius in a given circle; it is a unitless SI measure where $1$ radian is the angular distance covered when the arc length equals the radius.

Angular Position ($\theta$)

A measurement that describes the orientation of an object with respect to the x-axis, typically measured in degrees, radians, or revolutions.

Angular Displacement ($\Delta\theta$)

The change in angular position calculated as $\Delta\theta = \theta_f - \theta_i$, where positive values indicate counter-clockwise (CCW) rotation and negative values indicate clockwise (CW) rotation.

Average Angular Velocity ($\omega_{ave}$)

The rate of change of angular displacement over time, expressed as $\omega_{ave} = \frac{\Delta\theta}{\Delta t}$ with units such as $rad/s$ or $rev/min$ (rpm).

Instantaneous Angular Velocity ($\omega$)

The angular velocity of an object measured at any specific instant.

Average Angular Acceleration ($\alpha_{ave}$)

The rate of change of angular velocity over time, expressed as $\alpha_{ave} = \frac{\Delta\omega}{\Delta t}$ with units such as $rad/s^2$.

Instantaneous Angular Acceleration ($\alpha$)

The angular acceleration of an object measured at any specific instant.

Translational Speed ($v$) relationship

The link between translational and rotational speed of a point on a disk expressed as $v = r\omega$, where r is the distance from the axis of rotation and $\omega$ is in $rad/time$.

Translational Acceleration ($a_t$) relationship

The link between translational and rotational acceleration expressed as $a_t = r\alpha$, where $\alpha$ must be in $rad/time^2$.

Moment of Inertia ($I$)

A scalar measure of an object's resistance to rotational acceleration defined as $I = \sum m_i r_i^2$, where $r_i$ is the distance from each mass piece to the axis of rotation; the SI unit is $kg \cdot m^2$.

Rotational Kinetic Energy

The kinetic energy of a rigid rotating object, calculated using the formula $KE_{rot} = \frac{1}{2}I\omega^2$.

Thin hoop, radius R (Moment of Inertia)

For an axis through the center, the moment of inertia is $I = MR^2$.

Solid cylinder, radius R (Moment of Inertia)

For an axis through the center, the moment of inertia is $I = \frac{1}{2}MR^2$.

Uniform sphere, radius R (Moment of Inertia)

For an axis through the center, the moment of inertia is $I = \frac{2}{5}MR^2$.

Rolling Without Slipping (Velocity)

A condition where the translational speed of the center of mass is related to the rotational speed by the formula $v_{cm} = R\omega$.

Work Done by Torque

For a constant torque applied through an angle, the work is calculated as $W = \tau\theta$.

Power ($P$) in Rotational Motion

The rate of doing work related to torque and angular velocity, expressed as $P = \tau\omega$, which is analogous to $P = Fv$.

Torque ($\tau$)

A measure of the effectiveness of a force in producing rotation, defined by the product of the force and the lever arm as $\tau = rF\sin(\theta)$, with SI units of $m \cdot N$.

Lever Arm (Moment Arm)

The perpendicular distance from the axis of rotation to the line along which the force acts.

Net Torque ($\sum\tau$)

The sum of all torques due to all forces acting on an object.

Newton's 2nd Law in Rotational Form

The fundamental relationship stating that the net torque on an object is equal to its moment of inertia multiplied by its angular acceleration, or $\sum\tau = I\alpha$.

Angular Momentum ($L$)

The tendency of a spinning object to remain spinning in the same plane, calculated as $L = I\omega$ with units of $kg \cdot m^2/s$.

Conservation of Angular Momentum

A principle stating that as long as no external torques act on a system, the total angular momentum remains constant ($L_i = L_f$ or $I_i\omega_i = I_f\omega_f$).

Right-Hand Rule

A convention used for angular velocity where curling the fingers of the right hand in the direction of rotation results in the thumb pointing in the direction of the pseudovector.