Dynamics of Rotational Motion – Study Notes

Dynamics of Rotational Motion

Learning Outcomes

• Understand the concept of torque produced by a force.

• Analyze how net torque affects an object's rotational motion.

• Examine the motion of objects that both rotate and translate.

• Solve problems involving work and power in rotational systems.

• Apply the principle of conservation of angular momentum.

Torque

Definition and Calculation

Torque is a measure of the tendency of a force to rotate an object about an axis.

• Line of action: The straight line along which the force vector acts.

• Lever arm (l): The perpendicular distance from the axis of rotation (O) to the line of action of the force.

• Torque formula:

$\tau = F l$

• Alternatively, torque can be calculated as:

$\tau = F r \sin \phi = F_{tan} r$

• Where $\phi$ is the angle between the force and the position vector from the axis.

Example: Applying a force farther from the axis of a wrench is more effective in loosening a bolt due to a larger lever arm, resulting in greater torque.

Torque as a Vector

• Torque is a vector quantity, given by the cross product:

$\vec{\tau} = \vec{r} \times \vec{F}$

• The direction of $\vec{\tau}$ is determined by the right-hand rule.

Rotational Dynamics

Newton's Second Law for Rotation

The rotational analog of Newton's second law for a rigid body is:

$\sum \tau_z = I \alpha_z$

• $I$ is the moment of inertia about the rotation axis.

• $\alpha_z$ is the angular acceleration about the axis.

Example: Loosening or tightening a screw requires applying a torque to produce angular acceleration.

Problem-Solving Strategy for Rotational Dynamics

1. Sketch the situation and identify the object(s) and rotation axis.

2. Draw a free-body diagram, labeling all forces, dimensions, and angles.

3. Choose coordinate axes and indicate the positive sense of rotation (clockwise or counterclockwise).

Rigid Body Rotation About a Moving Axis

Kinetic Energy of a Rotating and Translating Body

The total kinetic energy ($K$) of a rigid body that both translates and rotates is:

$K = \frac{1}{2} M v_{cm}^2 + \frac{1}{2} I_{cm} \omega^2$

• $M$ is the mass, $v_{cm}$ is the velocity of the center of mass, $I_{cm}$ is the moment of inertia about the center of mass, and $\omega$ is the angular velocity.

Example: A baton tossed in the air exhibits both translation of its center of mass and rotation about its center of mass.

Rolling Motion

Rolling Without Slipping

When an object rolls without slipping, its motion is a combination of translation and rotation.

• The condition for rolling without slipping is:

$v_{cm} = R \omega$

• $v_{cm}$ is the velocity of the center of mass, $R$ is the radius, and $\omega$ is the angular velocity.

Rolling with Slipping

• If $v_{cm} \neq R \omega$, the object is slipping as it rolls (e.g., drag racer tires spinning and producing smoke).

Combined Translation and Rotation: Dynamics

• The acceleration of the center of mass is given by:

$\sum \vec{F}_{ext} = M \vec{a}_{cm}$

• The rotational motion about the center of mass is described by:

$\sum \tau_z = I_{cm} \alpha_z$

• These equations are valid if the axis through the center of mass is an axis of symmetry and does not change direction.

Rolling Friction

• Rolling friction can be neglected if both the rolling object and the surface are perfectly rigid.

• If either deforms, mechanical energy is lost, slowing the motion.

Work and Power in Rotational Motion

Work Done by a Torque

• A tangential force applied to a rotating object does work:

$W = \int_{\theta_i}^{\theta_f} \tau_z d\theta$

• The total work done by torque equals the change in rotational kinetic energy.

Power Due to a Torque

• The power delivered by a torque is:

$P = \tau_z \omega$

• Example: For a helicopter rotor spinning at constant rate, engine work balances air resistance, so net work and kinetic energy remain constant.

Angular Momentum

Definition and Calculation

• For a particle of mass $m$ at distance $r$ from the axis, rotating with angular speed $\omega$:

$L_i = m_i r_i^2 \omega$

• For a rigid body rotating about a symmetry axis:

$\vec{L} = I_{axis} \vec{\omega}$

• The direction of $\vec{L}$ is parallel to $\vec{\omega}$.

Angular Momentum for a System of Particles

• The rate of change of total angular momentum equals the sum of external torques:

$\sum \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$

Conservation of Angular Momentum

Principle

• If the net external torque on a system is zero, the total angular momentum is conserved:

$\vec{L}_{initial} = \vec{L}_{final}$

• Example: A spinning person pulling in their arms increases their rotation rate to conserve angular momentum.

• Example: A falling cat twists its body so that its total angular momentum remains zero, allowing it to land on its feet.

Gyroscopes and Precession

Gyroscope Motion

• For a gyroscope, the axis of rotation changes direction—a motion called precession.

• If a flywheel is not spinning, its initial angular momentum is zero; torque causes it to fall.

• If the flywheel is spinning, torque causes the angular momentum vector to change direction, not magnitude, resulting in precession around the pivot.

Example: The gyroscope's axis traces a circle as it precesses, maintaining its angular momentum magnitude but changing its direction.