Chapter 7: Rotational Motion – Physics with Algebra Study Notes

Rotational Motion

Describing Circular and Rotational Motion

Rotational motion refers to the movement of objects that spin about an axis. Understanding this motion is essential for analyzing systems ranging from wheels to planetary orbits.

• Rotational motion: The motion of an object spinning about a fixed axis.

• Circular motion: The motion of a particle along a circular path.

• Angular position (θ): The angle from the positive x-axis to the particle's location, measured in radians. Counterclockwise is positive, clockwise is negative.

• Arc length (s): The distance traveled along the circular path.

• Relationship: $\theta = \frac{s}{r}$ (radians), where r is the radius.

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

Angular Displacement and Angular Velocity

Angular displacement and angular velocity are the rotational analogs of linear displacement and velocity. They describe how much and how quickly an object rotates.

• Angular displacement (Δθ): The change in angular position.

• Angular velocity (ω): The rate of change of angular displacement, measured in rad/s.

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

• Angular speed: The absolute value of angular velocity.

• Period (T): The time for one complete revolution. $\omega = \frac{2\pi}{T}$

• Frequency (f): Number of revolutions per second. $\omega = 2\pi f$

Relating Linear and Angular Quantities

Linear and angular quantities are closely related in rotational motion. The speed of a point on a rotating object depends on its distance from the axis.

• Linear speed (v): $v = \omega r$

• Points farther from the axis move faster.

• Angular speed must be in rad/s for calculations.

Rotation of a Rigid Body

A rigid body is an object whose size and shape do not change as it moves. All points on a rigid body rotate with the same angular velocity, but their linear speeds vary with distance from the axis.

• Rigid body: Extended object with constant size and shape.

• All points have the same angular velocity.

• Linear speed increases with distance from the axis.

Angular Acceleration

Angular acceleration describes how quickly the angular velocity changes. It is the rotational analog of linear acceleration.

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

• Units: rad/s2

• Positive α: speeding up in the counterclockwise direction.

• Negative α: slowing down or speeding up in the clockwise direction.

Linear and Circular Motion Analogy

Variables and equations for linear motion have direct analogs in circular motion.

• Position: x (linear), θ (angular)

• Velocity: v (linear), ω (angular)

• Acceleration: a (linear), α (angular)

• Equations:

  • Linear: $\Delta x = v \Delta t + \frac{1}{2} a (\Delta t)^2$

  • Angular: $\Delta \theta = \omega \Delta t + \frac{1}{2} \alpha (\Delta t)^2$

Tangential Acceleration

Tangential acceleration is the component of acceleration directed tangent to the circle. It measures how quickly the speed around the circle increases.

• Tangential acceleration (at): $a_t = r \alpha$

• Relates angular acceleration to linear acceleration.

Torque

Torque is the rotational equivalent of force. It measures the ability of a force to cause rotation about an axis.

• Torque (τ): $\tau = r F \sin \phi$

• Depends on force magnitude, distance from pivot, and angle of application.

• Units: N·m (newton-meters)

• Moment arm: Perpendicular distance from the line of action to the pivot.

• Positive torque: Counterclockwise rotation; Negative torque: Clockwise rotation.

Center of Gravity and Gravitational Torque

The center of gravity is the point where the net force of gravity acts on an object. Gravitational torque can be calculated by considering the object's weight acting at this point.

• Center of gravity: Point where gravity acts as a single force.

• Gravitational torque: $\tau = r_{\perp} mg$

• Objects come to rest with the center of gravity below the pivot.

Moment of Inertia

Moment of inertia is the rotational equivalent of mass. It quantifies how mass is distributed relative to the axis of rotation and affects the angular acceleration produced by a given torque.

• Moment of inertia (I): $I = \sum m_i r_i^2$

• Depends on mass and its distribution.

• Units: kg·m2

• Common shapes:

  Object	Axis	Moment of Inertia (I)
  Thin rod	Center	$\frac{1}{12} M L^2$
  Thin rod	End	$\frac{1}{3} M L^2$
  Cylinder	Center	$\frac{1}{2} M R^2$
  Cylindrical hoop	Center	$M R^2$
  Solid sphere	Diameter	$\frac{2}{5} M R^2$
  Spherical shell	Diameter	$\frac{2}{3} M R^2$

Newton’s Second Law for Rotational Motion

Newton’s second law for rotation relates net torque to angular acceleration and moment of inertia.

• Formula: $\tau_{\text{net}} = I \alpha$

• Analogous to $F_{\text{net}} = m a$ in linear motion.

Rolling Motion

Rolling motion combines rotation and translation. For objects rolling without slipping, the linear speed of the center is linked to the angular speed.

• Rolling constraint: $v = \omega R$

• The point at the bottom of the wheel is instantaneously at rest.

Summary Table: Linear vs. Rotational Motion

Linear Motion	Rotational Motion
Position (x)	Angle (θ)
Velocity (v)	Angular velocity (ω)
Acceleration (a)	Angular acceleration (α)
Force (F)	Torque (τ)
Mass (m)	Moment of inertia (I)

Example Applications

• Calculating angular velocity and speed for rotating objects (e.g., wind turbine blades, car tires).

• Finding torque required to open doors or rotate objects.

• Balancing seesaws and determining center of gravity.

• Estimating angular acceleration for falling poles or spinning flywheels.

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