Rotational Motion, Angular Momentum, and Static Equilibrium

Rotational Motion and Torque

Definition and Concept of Torque

Torque (\( \tau \)) is a fundamental concept in rotational dynamics, representing the rotational equivalent of force. It measures the tendency of a force to rotate an object about an axis.

• Torque as a Vector: Torque is a vector quantity, meaning it has both magnitude and direction.

• Formula for Magnitude: The magnitude of torque produced by a force F applied at a position vector r from the axis of rotation, at an angle \( \theta \) between r and F:

$\tau = r F \sin \theta$

• Direction: Determined by the right-hand rule. Counterclockwise torques are considered positive, clockwise torques negative.

• Special Cases:

  • If r = 0 (force applied at the axis), then \( \tau = 0 \).

  • If \( \theta = 0 \) or \( 180^\circ \) (force along the line of r), then \( \tau = 0 \).

Example: Using a wrench to loosen a bolt: the farther from the axis (larger r), the greater the torque for the same force.

Moment of Inertia (Rotational Inertia)

Definition and Dependence

The moment of inertia (I) quantifies an object's resistance to changes in its rotational motion. It depends on both the mass and how that mass is distributed relative to the axis of rotation.

• For a point mass:

$I = m r^2$

• For extended bodies: Sum or integrate m r^2 over the entire mass distribution.

• Key Principle: The farther the mass is from the axis, the greater the moment of inertia.

Example: Comparing a solid sphere and a hollow spherical shell of the same mass and radius: the hollow shell has a larger moment of inertia because its mass is distributed farther from the axis.

Rolling Motion

Combined Translational and Rotational Kinetic Energy

When an object rolls without slipping, its motion is a combination of translation (movement of the center of mass) and rotation about its center of mass.

• Translational Kinetic Energy: $\frac{1}{2} M v^2$

• Rotational Kinetic Energy: $\frac{1}{2} I \omega^2$

• Total Kinetic Energy:

$K_{\text{total}} = \frac{1}{2} I \omega^2 + \frac{1}{2} M v^2$

• For rolling without slipping: $v = \omega R$

Example: A bicycle wheel rolling on the road has both rotational and translational kinetic energy.

Analogies Between Linear and Rotational Dynamics

Linear Motion	Rotational Motion
Displacement: x	Angular displacement: \( \theta \)
Velocity: v	Angular velocity: \( \omega \)
Acceleration: a	Angular acceleration: \( \alpha \)
Mass: m	Moment of inertia: I
Force: F	Torque: \( \tau \)
Momentum: p = m v	Angular momentum: L = I \omega
Newton's 2nd Law: F = m a	Rotational analog: \( \tau = I \alpha \)

Angular Momentum and Its Conservation

Definition and Conservation Law

Angular momentum (L) is the rotational analog of linear momentum. For a rigid body rotating about a fixed axis:

$L = I \omega$

• Conservation Law: If the net external torque on a system is zero, its total angular momentum remains constant.

$\sum \tau_{\text{ext}} = \frac{\Delta L}{\Delta t}$

If $\sum \tau_{\text{ext}} = 0$, then $L$ is constant.

• Application: When a system changes configuration (e.g., a figure skater pulling in arms), $I$ and $\omega$ can change, but their product remains constant:

$I_1 \omega_1 = I_2 \omega_2$

Example: A skater spins faster when pulling arms in (decreasing I, increasing \( \omega \)).

Sample Problem: Merry-Go-Round

• A child moves from the center to the rim of a rotating merry-go-round. The system's total angular momentum is conserved, so:

$I_1 \omega_1 = I_2 \omega_2$

• Calculate $I_1$ and $I_2$ by summing the moments of inertia of the merry-go-round and the child (as a point mass at the appropriate radius).

• Solve for the new angular velocity $\omega_2$.

Sample Problem: Stacked Rotating Disks

• When a stationary disk is dropped onto a rotating disk, they rotate together. The final angular velocity is found by conserving angular momentum:

$I_1 \omega_1 = (I_1 + I_2) \omega_2$

Solve for $\omega_2$.

Vector Nature of Angular Momentum and the Right-Hand Rule

Direction of Angular Quantities

• Angular velocity and angular momentum are vectors along the axis of rotation.

• Right-Hand Rule: Curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of the angular velocity vector.

• Counterclockwise rotation: vector points up; clockwise: vector points down.

Example: A person walking on a rotating platform causes the platform to rotate in the opposite direction to conserve angular momentum.

Static Equilibrium

Definition and Conditions

Static equilibrium occurs when an object is at rest and remains at rest. This requires both translational and rotational equilibrium.

• Translational Equilibrium: Net force on the object is zero.

• Rotational Equilibrium: Net torque on the object is zero about any axis.

Mathematical Conditions:

• $\sum F_x = 0$ (sum of forces in x-direction)

• $\sum F_y = 0$ (sum of forces in y-direction)

• $\sum \tau = 0$ (sum of torques about any point)

Sign Conventions:

• Forces to the right/up: positive; left/down: negative.

• Torques: counterclockwise positive, clockwise negative.

Example: A ladder leaning against a wall must satisfy both net force and net torque conditions to remain at rest.

Center of Gravity and Center of Mass

Definition and Determination

• Center of Gravity: The point where the total weight of an object acts. For uniform gravitational fields, it coincides with the center of mass.

• Finding the Center of Gravity: Suspend the object from different points; the intersection of the vertical lines from each suspension point locates the center of gravity.

Example: A dancer balancing on one foot is in static equilibrium when the line of action of the normal force and gravity passes through the center of gravity.

Torque Due to Gravity

• Gravity acts at the center of gravity (center of mass for uniform fields).

• To calculate torque due to gravity, use the perpendicular distance from the axis to the line of action of the weight.

$\tau = r_{\perp} F$

Summary Table: Static Equilibrium Conditions

Condition	Mathematical Expression	Physical Meaning
Translational Equilibrium (x)	$\sum F_x = 0$	No net force in x-direction
Translational Equilibrium (y)	$\sum F_y = 0$	No net force in y-direction
Rotational Equilibrium	$\sum \tau = 0$	No net torque about any axis

Key Problem-Solving Strategies

• Identify all forces acting on the system (gravity, normal, friction, applied forces).

• Choose a convenient axis for calculating torques (often where unknown forces act).

• Apply equilibrium conditions to solve for unknowns.

• Use the right-hand rule to determine the direction of torques and angular quantities.

Additional info: Some context and examples were expanded for clarity and completeness, including the analogy table and explicit problem-solving steps.