Newton's Law for Rotation, Kinematic Equations, and Angular Momentum Conservation

Newton's Law for Rotation and Kinematic Equations

Newton's Law for Rotation

Newton's Second Law, which describes the relationship between force, mass, and acceleration in linear motion, has a direct analog in rotational motion. In rotational dynamics, torque plays the role of force, and moment of inertia replaces mass.

• Force (F) causes translation; Torque (\(\tau\)) causes rotation.

• Mass (m) measures inertia; Moment of Inertia (I) measures rotational inertia.

• Linear acceleration (a) corresponds to angular acceleration (\(\alpha\)).

The equations are:

• Linear: $\vec{F}_{\text{net}} = m \vec{a}$

• Rotational: $\vec{\tau}_{\text{net}} = I_{\text{tot}} \vec{\alpha}$

Dimensional Check: $[\tau] = [I][\alpha] = (\text{kg} \cdot \text{m}^2)(\text{rad}/\text{s}^2) = \text{N} \cdot \text{m}$

Step-by-Step for Newton's Law for Rotation

1. Identify pivot/axis of rotation (P): Choose the point or axis about which rotation occurs.

2. Evaluate net torque (\(\tau_{\text{net}}\)):

   • Use the convention for the direction (e.g., counterclockwise positive).

   • Calculate torque for each force: $\tau = \pm rF \sin \beta$

   • Sum all torques to get the net torque.

3. Evaluate total moment of inertia (\(I_{\text{tot}}\)):

   • Identify the center of mass (COM) axis for each mass.

   • Sum the moments of inertia for all masses.

   • If the axis is not at the COM, use the parallel axis theorem: $I_P = I_{\text{COM}} + Mh^2$

4. Combine to solve for the desired quantity: $\vec{\alpha} = \frac{\vec{\tau}_{\text{net}}}{I_{\text{tot}}}$

Kinematic Equations for Rotational Motion (Constant Angular Acceleration)

When angular acceleration is constant, the following kinematic equations apply (analogous to linear kinematics):

Equation	Variables
$\omega_f = \omega_i + \alpha t$	Final angular velocity, initial angular velocity, angular acceleration, time
$\theta_f - \theta_i = \omega_i t + \frac{1}{2} \alpha t^2$	Angular displacement, initial angular velocity, angular acceleration, time
$\omega_f^2 = \omega_i^2 + 2\alpha (\theta_f - \theta_i)$	Final and initial angular velocity, angular acceleration, angular displacement

• \(\omega\): Angular velocity

• \(\alpha\): Angular acceleration

• \(\theta\): Angular position

Key Steps:

1. Visualize motion and set coordinate system and convention.

2. Identify all known and unknown variables.

3. Choose the appropriate kinematic equation.

4. Plug in values and solve for the unknown.

Examples and Applications

• Bicycle Wheel: Calculating angular acceleration and revolutions when a force is applied tangentially to the rim.

• Merry-Go-Round: Finding angular acceleration, angular velocity after a certain number of revolutions, and the time taken.

• Pivoted Rod: Evaluating angular acceleration and velocity after rotation by a given angle under multiple forces.

Angular Momentum and Its Conservation

Angular Momentum by Analogy

Angular momentum is the rotational analog of linear momentum. Just as force is the rate of change of linear momentum, torque is the rate of change of angular momentum.

• Linear: $\vec{F}_{\text{net}} = \frac{\Delta \vec{p}}{\Delta t}$

• Rotational: $\vec{\tau}_{\text{net}} = \frac{\Delta \vec{L}}{\Delta t}$

• \(\vec{L}\): Angular momentum

Angular Momentum of a Point Mass

• Definition: $\vec{L} = \vec{r} \times \vec{p}$, where $\vec{p} = m\vec{v}$

• Magnitude: $L = \pm r p \sin \beta$

• Direction: Right-hand rule: curl fingers from $\vec{r}$ to $\vec{p}$, thumb points in direction of $\vec{L}$.

Angular Momentum of a Rotating Point Mass

• For a mass $m$ moving in a circle of radius $R$ with angular velocity $\omega$:

• $L = mR^2 \omega = I_P \omega$

General Formula for Angular Momentum about Rotation/Pivot Axis

• $\vec{L}_P = I_P \vec{\omega}$

• This formula applies for any axis; use the parallel axis theorem if needed.

Conservation of Angular Momentum

Angular momentum is conserved if the net external torque on a system is zero.

• Statement: $\vec{L}_{\text{total, initial}} = \vec{L}_{\text{total, final}}$ if $\vec{\tau}_{\text{net}} = 0$

• Applies to both point masses and rigid bodies.

Examples and Applications

• Planetary Motion: Conservation of angular momentum explains Kepler's second law (equal areas in equal times).

• Figure Skater: Spins faster when arms are pulled in due to reduced moment of inertia.

• Rotating Chair: A person holding a spinning wheel can change their rotation by reorienting the wheel, demonstrating conservation of angular momentum.

• Bullet and Disk: When a bullet embeds in a disk, the total angular momentum before and after the collision is conserved (if no external torque acts).

• Carousel and Kid: As a kid walks toward the center, the system's moment of inertia decreases, so angular velocity increases to conserve angular momentum.

Steps to Apply Angular Momentum Conservation

1. Check if net external torque is zero along the axis of interest.

2. Write the conservation equation: Total Initial Angular Momentum = Total Final Angular Momentum

3. Identify knowns and unknowns, solve for the unknown.

Summary Tables

Linear Motion	Rotational Motion
Force (F)	Torque (\(\tau\))
Mass (m)	Moment of Inertia (I)
Acceleration (a)	Angular Acceleration (\(\alpha\))
Momentum (p)	Angular Momentum (L)
$\vec{F}_{\text{net}} = m\vec{a}$	$\vec{\tau}_{\text{net}} = I\vec{\alpha}$
$\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$	$\vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt}$

Conservation Law	Condition	Equation
Linear Momentum	No net external force	$\vec{p}_{\text{total, i}} = \vec{p}_{\text{total, f}}$
Angular Momentum	No net external torque	$\vec{L}_{\text{total, i}} = \vec{L}_{\text{total, f}}$

Additional info: These notes include step-by-step problem-solving strategies, worked examples, and analogies between linear and rotational motion to reinforce conceptual understanding. The content is suitable for exam preparation in introductory college physics courses covering rotational dynamics and angular momentum.