Rotational Equilibrium and Rotational Dynamics

Flashcards covering key definitions, equations, and principles of rotational kinematics and dynamics from Grade 12 General Physics 1 Module 1.

What is rotational kinematics?

The description of rotational motion without regard to force or mass, describing the relationships among rotation angle ($\theta$), angular velocity ($\omega$), angular acceleration ($\alpha$), and time ($t$).

How is a rigid body defined in the context of rotational motion?

An object that keeps its shape and is not deformed in any way by its motion.

What is the formula for angular displacement ($\theta$)?

$\theta = \frac{s}{r}$, where $s$ is the arc length and $r$ is the radius.

How many radians are swept out during one complete revolution of $360^\circ$?

$2\pi\,\text{radians}$ (or approximately $6.28\,\text{radians}$).

What is the equivalent value of $1\,\text{radian}$ in degrees?

$$57.3^\circ$$

What is the relationship between tangential velocity ($v$), angular velocity ($\omega$), and radius ($r$)?

$$v = r\omega$$

What is the formula for average angular acceleration ($\alpha$)?

$$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega_2 - \omega_1}{t_2 - t_1}$$

What are the mathematical expressions for tangential acceleration ($a_{\text{tan}}$) and radial acceleration ($a_{\text{rad}}$)?

Tangential acceleration is $a_{\text{tan}} = r\alpha$ and radial acceleration is $a_{\text{rad}} = \omega^2 r$.

What standard sign convention is used for rotational direction?

Counterclockwise motion has a positive ($+$) value, while clockwise motion has a negative ($-$) value.

What is torque ($\tau$) and how is its magnitude calculated?

Torque (or Moment of Force) is the cross product of position vector and force vector that causes rotation; its magnitude is given by $\tau = r F \sin(\theta)$.

What three factors influence the magnitude of a torque exerted on an object?

1) The magnitude of the applied force, 2) The lever arm (moment arm), and 3) The angle between the force vector and the lever arm.

What two conditions must be satisfied for a system to maintain static equilibrium?

The net force acting on the object must be zero ($\sum \vec{F} = 0$) and the net torque acting on the object must be zero ($\sum \tau = 0$).

What is the rotational analog of Newton's Second Law of Motion?

$\tau = I\alpha$, where $\tau$ is torque, $I$ is moment of inertia, and $\alpha$ is angular acceleration.

How is work done by a torque calculated?

$W = \tau \theta$, where $\tau$ is torque and $\theta$ is angular displacement.

What is the formula for rotational kinetic energy?

$KE_{\text{rot}} = \frac{1}{2} I \omega^2$, where $I$ is moment of inertia and $\omega$ is angular velocity.

What is the total kinetic energy equation for a body undergoing both translation and rotation?

$$KE_{\text{total}} = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$$

What is the formula for angular momentum ($L$) of a rotating body?

$L = I\omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.

What does the Law of Conservation of Angular Momentum state?

The momentum of a system will not change unless an external torque is applied ($L_f = L_i$).