Circular and Rotational Motion: Study Notes

Circular Motion

Arc Length and Distance Along a Circle

Circular motion involves objects moving along a circular path. The arc length is the distance traveled along the circumference of a circle.

• Arc Length (S): The distance along the circle's edge, given by $S = r\theta$ where $r$ is the radius and $\theta$ is the angle in radians.

• Important: Always use radians (not degrees) in the formula for arc length.

• Conversion: $1\ \text{rad} = 57.3\ \text{degrees}$

Angular Displacement

Angular displacement measures the change in angle as an object moves along a circular path.

• Symbol: $\theta$

• Units: Radians (rad)

Angular Speed and Angular Acceleration

Angular speed and angular acceleration describe how quickly an object rotates and how its rotation rate changes.

• Angular Speed ($\omega$): $\omega = \frac{\Delta\theta}{\Delta t}$

• Angular Acceleration ($\alpha$): $\alpha = \frac{\Delta\omega}{\Delta t}$

Non-Uniform Circular Motion

When the speed of an object in circular motion changes, it experiences both tangential and centripetal accelerations.

• Tangential Acceleration ($a_t$): Changes the speed along the tangent to the circle.

• Centripetal Acceleration ($a_c$): Points toward the center, keeps the object moving in a circle.

• Total Acceleration: $a_{\text{total}} = \sqrt{a_t^2 + a_c^2}$

Examples

• Example: If a car moves in a circle of radius $10.25\ \text{m}$ in $2\ \text{min}$, its angular speed is $\omega = \frac{\Delta\theta}{\Delta t}$.

• Example: For a mass $m = 1300\ \text{kg}$, with $\omega = 0.67\ \text{rad/s}$, centripetal acceleration is $a_c = r\omega^2$.

Rotational Motion of Solid Objects

Rotation vs. Translation

Solid objects can rotate about an axis, and their rotational motion is described by angular quantities.

• Moment of Inertia ($I$): Analogous to mass in linear motion; measures resistance to rotational acceleration.

• Torque ($\tau$): Analogous to force; causes angular acceleration.

• Angular Displacement ($\theta$): Measures the angle rotated.

Torque and Net Torque

Torque is the rotational equivalent of force, calculated as:

• $\tau = rF\sin\theta$

• Net Torque ($\tau_{\text{net}}$): $\tau_{\text{net}} = I\alpha$ (Newton's Second Law for rotation)

• Direction: Torque points down is negative.

Center of Gravity

The center of gravity is the point where the object's mass is balanced.

• If held at the center of gravity, the object will not have angular acceleration; net torque is zero.

• To solve problems, set the sum of torques equal to zero for equilibrium.

• For discrete masses, center of gravity is found by $x_{cg} = \frac{\sum m_ix_i}{\sum m_i}$

Moment of Inertia

Moment of inertia depends on the mass distribution relative to the axis of rotation.

• Point Mass: $I = mr^2$

• Solid Disk: $I = \frac{1}{2}MR^2$

• Shell: $I = MR^2$

• Parallel Axis Theorem: $I = I_{cm} + Mh^2$ (where $h$ is the distance from the center of mass axis)

Examples

• Example: For a compact disk, $I = \frac{1}{2}MR^2$

• Example: For a shell, $I = MR^2$

Rolling Motion

Rolling: Translation and Rotation

Rolling motion combines both translation and rotation. The velocity of the center of mass and the angular velocity are related.

• Velocity of Center of Mass ($v_{cm}$): $v_{cm} = \omega R$

• Rolling without slipping: $v = \omega R$

• Combined motion: The total velocity at a point is the sum of translational and rotational velocities.

Example: Falling Bucket Problem

• Analyzing a bucket falling and rotating involves both linear and angular equations.

• Use $a = \alpha r$ to relate angular and linear acceleration.

Summary Table: Moments of Inertia for Common Shapes

Shape	Moment of Inertia ($I$)
Point Mass	$I = mr^2$
Solid Disk	$I = \frac{1}{2}MR^2$
Thin Shell	$I = MR^2$
Rod (about center)	$I = \frac{1}{12}ML^2$
Rod (about end)	$I = \frac{1}{3}ML^2$
Parallel Axis Theorem	$I = I_{cm} + Mh^2$
Additional info: Other shapes can be calculated using integration or standard formulas.

Key Formulas

• Arc Length: $S = r\theta$

• Angular Speed: $\omega = \frac{\Delta\theta}{\Delta t}$

• Angular Acceleration: $\alpha = \frac{\Delta\omega}{\Delta t}$

• Centripetal Acceleration: $a_c = r\omega^2$

• Torque: $\tau = rF\sin\theta$

• Newton's Second Law for Rotation: $\tau_{\text{net}} = I\alpha$

• Rolling Motion: $v = \omega R$

Additional info: These notes cover the essentials of circular and rotational motion, including the relationships between linear and angular quantities, and the calculation of moments of inertia for various shapes. For exam preparation, focus on understanding the physical meaning of each formula and how to apply them to solve problems involving rotation and rolling.