Rotational Motion

Describing Circular and Rotational Motion

Rotational motion refers to the movement of objects that spin about an axis. This type of motion is fundamental in understanding the behavior of many physical systems, from wheels to planetary orbits.

• Rotational motion: The object spins about a fixed axis.

• Circular motion: Every point on the object moves in a circle around the axis.

• Combination motion: An object rotates as it moves along a trajectory.

Angular Position

The angular position of a particle in circular motion is described by the angle θ, measured from the positive x-axis. The angle is positive when measured counterclockwise and negative when measured clockwise.

• Angle θ: Describes the particle's location in radians.

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

• Radius (r): The distance from the center of circular motion to the particle.

The relationship between arc length and angular position is given by:

One revolution corresponds to an angle of radians, and conversion factors among revolutions, radians, and degrees are essential for calculations.

Angular Displacement and Angular Velocity

Angular displacement is the change in angular position, and angular velocity is the rate at which this displacement occurs. For uniform circular motion, angular velocity is constant.

• Angular velocity (ω): The angular displacement per unit time.

• Angular speed: The absolute value of angular velocity.

• Period (T): The time for one complete revolution.

• Frequency (f): Number of revolutions per second.

Key equations:

Relating Speed and Angular Speed

Points farther from the axis of rotation move at higher linear speeds. The linear speed (v) at any point is related to the angular speed (ω) by:

The Rotation of a Rigid Body

Rigid Body Rotation

In a rigid body, every point has the same angular velocity, but points at different distances from the axis have different linear speeds.

Angular Acceleration

Angular acceleration (α) is the rate of change of angular velocity. It is measured in radians per second squared (rad/s²).

Synthesis: Linear and Circular Motion

Variables and equations for linear motion have direct analogs in circular motion. This synthesis helps in understanding rotational dynamics using familiar concepts from linear motion.

Tangential Acceleration

Tangential acceleration is the component of acceleration directed along the tangent to the circle. It measures the rate at which the speed around the circle increases.

Torque

Definition and Calculation of Torque

Torque is the rotational equivalent of force. It depends on the magnitude of the force, the distance from the pivot, and the angle at which the force is applied.

• Torque (τ):

• Moment arm: The perpendicular distance from the pivot to the line of action of the force.

Alternate expression for torque:

Torque is positive for counterclockwise rotation and negative for clockwise rotation.

Net Torque

The net torque is the sum of all individual torques acting on an object:

Gravitational Torque and the Center of Gravity

Gravitational Torque

Gravity exerts a force and a torque on every particle of an object. The net gravitational torque can be calculated by assuming the weight acts at the center of gravity.

Example: Gravitational Torque on a Flagpole

For a flagpole extending from a wall, the gravitational torque about the point of attachment is calculated using the moment arm and the weight force.

Calculating the Center of Gravity

The center of gravity is the point where the net torque due to gravity is zero. For a system of point masses:

Example: Balancing a Seesaw

To balance a seesaw, the combined center of gravity of the children must be at the pivot point. The heavier child sits closer to the pivot.

Rotational Dynamics and Moment of Inertia

Moment of Inertia

Moment of inertia (I) is the rotational equivalent of mass. It depends on both the mass and its distribution relative to the axis of rotation.

Newton’s Second Law for Rotational Motion

Newton’s second law for rotation states that a net torque causes angular acceleration:

Moments of Inertia of Common Shapes

Different shapes have characteristic moments of inertia, depending on their mass and geometry.

Rolling Motion

Rolling Motion and Constraints

Rolling motion is a combination of rotation and translation. For objects rolling without slipping, the velocity of the center is linked to the angular velocity:

The point at the bottom of a rolling object is instantaneously at rest due to the cancellation of translational and rotational velocities.

Example: Rotating Your Tires

When driving, the tires rotate many times per second. The speed of a point at the top of the tire is twice the speed of the car.

Key equations: