Dynamics of Rotational Motion: Torque, Angular Momentum, and Equilibrium

Dynamics of Rotational Motion

Introduction

This section explores the fundamental principles governing rotational motion, focusing on torque, angular acceleration, angular momentum, and the equilibrium of rigid bodies. These concepts are essential for understanding the behavior of rotating systems in physics.

Torque

Definition and Calculation

• Torque (\(\tau\)) is the rotational equivalent of force. It measures the tendency of a force to rotate an object about an axis.

• Mathematically, torque is defined as the product of the force applied and the moment arm (the perpendicular distance from the axis of rotation to the line of action of the force): \[ \tau = F l \] where \(F\) is the force and \(l\) is the moment arm.

• Units: [N·m] (Newton-meters)

• Three equivalent ways to calculate torque: \[ \tau = F l = r F \sin \phi = F_{\text{tan}} r \] where \(\phi\) is the angle between the force and the lever arm, and \(F_{\text{tan}}\) is the tangential component of the force.

Sign Convention

• Torque is a vector quantity, similar to angular velocity and acceleration.

• By convention:

  • Positive torque (+): Counterclockwise rotation

  • Negative torque (−): Clockwise rotation

Rotational Newton's Second Law

Relationship Between Torque and Angular Acceleration

• Newton's Second Law for rotation relates the net torque to the angular acceleration (\(\alpha\)) and the moment of inertia (\(I\)): \[ \sum \tau = I \alpha \] where \(\sum \tau\) is the sum of all torques, \(I\) is the moment of inertia, and \(\alpha\) is the angular acceleration.

• This is analogous to \(F = ma\) for linear motion.

Work and Power in Rotational Motion

Rotational Work

• If torque is constant and the angle changes, the rotational work done is: \[ W = \tau \Delta \theta \] where \(\Delta \theta\) is the angular displacement (in radians).

Rotational Power

• Power in rotational motion is given by: \[ P = \frac{W}{\Delta t} = \tau \frac{\Delta \theta}{\Delta t} = \tau \omega \] where \(\omega\) is the angular velocity.

Angular Momentum

Definition and Calculation

• The angular momentum (\(L\)) of a rigid body is: \[ L = I \omega \] where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

• Units: [N·m·s]

• For a single particle: \[ L = m v l \] where \(m\) is mass, \(v\) is velocity, and \(l\) is the perpendicular distance to the axis of rotation.

Rate of Change of Angular Momentum

• The rate of change of angular momentum is equal to the net external torque: \[ \sum \tau = \frac{\Delta L}{\Delta t} \]

Conservation of Angular Momentum

Principle

• If the sum of the external torques acting on a system is zero, the total angular momentum remains constant (is conserved): \[ L_f = L_i \] \[ I_f \omega_f = I_i \omega_i \] where subscripts \(i\) and \(f\) refer to initial and final states, respectively.

Example Application

• A figure skater pulling in their arms to spin faster demonstrates conservation of angular momentum: as \(I\) decreases, \(\omega\) increases to keep \(L\) constant.

Equilibrium of Rigid Bodies

Conditions for Equilibrium

• For a rigid body to be in equilibrium, both the net force and the net torque must be zero: \[ \sum F_x = 0 \] \[ \sum F_y = 0 \] \[ \sum \tau = 0 \]

Example Application

• Balancing a seesaw or carrying a heavy object with two people involves ensuring both force and torque equilibrium.

Right Hand Rule

Direction of Angular Vectors

• The right hand rule determines the direction of angular velocity, angular momentum, and torque vectors.

• To use the rule: curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of the vector.

Summary Table: Key Rotational Quantities

Quantity	Symbol	Formula	SI Unit
Torque	\(\tau\)	\(\tau = F l\)	N·m
Moment of Inertia	\(I\)	Depends on mass distribution	kg·m2
Angular Acceleration	\(\alpha\)	\(\alpha = \frac{\tau}{I}\)	rad/s2
Angular Momentum	\(L\)	\(L = I \omega\)	N·m·s
Rotational Work	\(W\)	\(W = \tau \Delta \theta\)	J (Joules)
Rotational Power	\(P\)	\(P = \tau \omega\)	W (Watts)

Examples

• Flywheel Problem: Given a flywheel with mass 300 kg, moment of inertia 580 kg·m2, and a constant torque of 2000 N·m, calculate:

  • Angular acceleration: \(\alpha = \frac{\tau}{I} = \frac{2000}{580} \approx 3.45\ \text{rad/s}^2\)

  • Angular velocity after 4 revolutions: \(\omega^2 = 2 \alpha \theta\), where \(\theta = 4 \times 2\pi\) radians

  • Work done: \(W = \tau \Delta \theta\)

• Seesaw Equilibrium: To balance a seesaw, set the sum of torques about the pivot to zero and solve for the pivot position.

Additional info: These notes expand on the provided slides by including definitions, formulas, and example calculations for clarity and completeness.