AP Physics 1- 5.2 Connecting Linear and Rotational Motion- Study Notes- New Syllabus

Connecting Linear and Rotational Motion

Linear and rotational motions are closely related. For a point on the rim of a rotating body (radius \( r \)), linear quantities can be expressed in terms of angular quantities.

 Angular Quantities for a Rigid System

For a rigid system, all points have the same angular displacement, angular velocity, and angular acceleration, regardless of their distance from the axis of rotation.

However, their linear velocities and linear accelerations depend on their distance \( r \) from the axis.

Displacement

The arc length \( s \) traveled by a point on the rim of a rotating object is related to angular displacement \( \theta \) by:

\( s = r \theta \)

Units: meters (m).

Velocity

The linear velocity \( v \) of a point on the rim is proportional to angular velocity \( \omega \):

\( v = r \omega \)

Direction of \( v \): Tangent to the circular path (perpendicular to the radius).

Acceleration

There are two components of linear acceleration for a rotating body:

• Tangential acceleration: \( a_t = r \alpha \) (due to angular acceleration).
• Centripetal acceleration: \( a_c = \dfrac{v^2}{r} = r \omega^2 \) (directed toward the center of rotation).

Kinematic Analogies

The equations of rotational motion mirror those of linear motion:

• Linear: \( v = v_0 + a t \)    ↔    Rotational: \( \omega = \omega_0 + \alpha t \)
• Linear: \( s = v_0 t + \dfrac{1}{2} a t^2 \)    ↔    Rotational: \( \theta = \omega_0 t + \dfrac{1}{2} \alpha t^2 \)
• Linear: \( v^2 = v_0^2 + 2 a s \)    ↔    Rotational: \( \omega^2 = \omega_0^2 + 2 \alpha \theta \)