Physics: Angular and Linear Velocity

Study Notes

Circular Motion Concepts

Velocity in circular motion can be expressed as ( v = \omega r ), where ( \omega ) is the angular velocity in radians per second and ( r ) is the radius.
The arc length traveled in one second equals the product of angular velocity and radius.

Centripetal Force

Centripetal force is the resultant force acting on an object in uniform circular motion, directed towards the center of the circle.
Formula for centripetal force: ( F = \frac{mv^2}{r} ) or ( F = mw^2 r ), where ( m ) is mass, ( v ) is velocity, and ( r ) is the radius of the circle.

Centripetal Acceleration

Centripetal acceleration is always directed towards the center of the circle, proportional to the resultant force according to Newton’s second law.
Formula for centripetal acceleration: ( a = \frac{v^2}{r} ) or ( a = w^2 r ).

Definitions

Period of Rotation: Time taken for one complete revolution in circular motion.
Frequency: Number of complete rotations or oscillations per unit time.
Radian: An angle measure where one radian is the angle subtended by an arc length equal to the radius.

Angular Conversion

There are ( 2\pi ) radians in a complete circle (360°).
To convert radians to degrees: ( \text{degrees} = \frac{\text{radians} \times 360}{2\pi} ).
To convert degrees to radians: ( \text{radians} = \frac{\text{degrees} \times 2\pi}{360} ).

Angular Velocity

Angular velocity (( \omega )) measures the rate of change of angular displacement and is a vector quantity.
Its sign indicates direction (positive for counter-clockwise, negative for clockwise).
In simple harmonic motion, the relation between angular velocity and frequency is given by ( \omega = 2\pi f ).
Circular motion exhibits simple harmonic motion characteristics when viewed from the side, illustrating a connection between the two concepts.

Description

Explore the relationship between linear velocity and angular velocity through the equation v = ωr. This quiz covers the basic geometric principles of circles and how they define motion in terms of both angular and linear measurements.