Center of Mass, Rotational Motion, and Rolling Dynamics

Center of Mass (CM) & Center of Gravity (CG)

Definition and Concepts

The center of mass (CM) is the point where the mass of a system can be considered to be concentrated for the purposes of analyzing translational motion. The center of gravity (CG) is the point where the gravitational force can be considered to act.

• For uniform gravitational fields, CM and CG coincide.

Formulas (Discrete Particles)

• Center of mass coordinates:

• $x_{CM} = \frac{\sum m_i x_i}{\sum m_i}$

• $y_{CM} = \frac{\sum m_i y_i}{\sum m_i}$

• $z_{CM} = \frac{\sum m_i z_i}{\sum m_i}$

Two-Particle System

• $x_{CM} = \frac{m_A x_A + m_B x_B}{m_A + m_B}$

Properties of Center of Mass

• CM follows the same path as a point particle under net external force.

• For uniform symmetric objects, CM = geometric center.

• CM/CG can lie outside the object (e.g., donut hole, high jumper's body).

• Human body CM: Approx. 55–60% of height above the ground (varies with posture).

Angular Quantities

Angular Displacement

• $\theta = \frac{l}{r}$ (units: radians)

Conversions

• $1\ \text{rev} = 2\pi\ \text{rad} = 360^\circ$

Arc Length

• $l = r\theta$

Angular Velocity

• $\omega = \frac{\Delta \theta}{\Delta t}$ (units: rad/s)

Angular Acceleration

• $\alpha = \frac{\Delta \omega}{\Delta t}$ (units: rad/s$^2$)

Linear and Angular Relations

• $v = \omega r$

• $a_{\text{tan}} = \alpha r$

Rolling Motion

Rolling Without Slipping

• $v_{CM} = R\omega$

• $a_{CM} = R\alpha$

Kinetic Energy of a Rolling Object

• $KE = \frac{1}{2} M v_{CM}^2 + \frac{1}{2} I \omega^2$

• Note: Objects with smaller $I/MR^2$ reach the bottom of an incline faster.

Torque

Definition

Torque is the rotational effect of a force, causing an object to rotate about an axis.

Formulas

• $\tau = r F \sin \theta$

• $\tau = F \cdot \ell$

• Where $\ell$ is the perpendicular lever arm.

• Units: N·m

Sign Convention

• Counterclockwise (CCW) = positive

• Clockwise (CW) = negative

Rotational Dynamics

Newton's 2nd Law (Rotational Form)

• $\tau_{\text{net}} = I \alpha$

Moment of Inertia (Examples)

Object	Moment of Inertia ($I$)
Hoop (about center)	$I = MR^2$
Solid cylinder/disc	$I = \frac{1}{2}MR^2$
Sphere	$I = \frac{2}{5}MR^2$
Rod (center)	$I = \frac{1}{12}ML^2$
Rod (end)	$I = \frac{1}{3}ML^2$

Rotational Kinetic Energy

• $KE_{rot} = \frac{1}{2} I \omega^2$

Total Energy (Rolling)

• $KE_{total} = \frac{1}{2} M v_{CM}^2 + \frac{1}{2} I \omega^2$

Linear & Rotational Analogy

Many equations in rotational motion have direct analogs in linear motion, such as force/torque, mass/moment of inertia, and acceleration/angular acceleration. Understanding these analogies helps in solving rotational dynamics problems.