Circular Motion and Gravitation · 8 min read
Angular velocity and centripetal force
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Introduction to Circular Motion
Imagine you are tieing a stone to a piece of string and swinging it over your head. The stone is moving in a circle. Even if you don't speed up, the stone is constantly changing direction. In Physics, because the direction is changing, we say the stone is accelerating.
Angular Displacement
When an object moves in a circle, we don't just measure the distance in meters. We also look at the angle it has covered. If an okada wheel turns half-way, it has covered 180 degrees or pi radians.
Simply put, it is the 'angle distance' an object covers while moving in a circle.
Angular Velocity
Think of a ceiling fan. When you turn it to 'Speed 5', it rotates very fast. The rate at which it covers those angles is called angular velocity. We use the Greek letter omega (w) to represent it.
It tells us how fast an object is spinning or rotating every second.
Relationship between Linear and Angular Velocity
If you are standing on a rotating merry-go-round, the person at the edge moves faster in a straight line than the person near the center, even though they have the same angular velocity. The formula connecting them is: v = rw, where v is linear velocity, r is radius, and w is angular velocity.
Centripetal Force
To keep the stone moving in a circle, you must pull the string inward. If the string breaks, the stone flies off. That 'inward pull' is what we call centripetal force. Without it, circular motion is impossible.
It is the center-seeking force that keeps an object from flying away from its circular path.
Centripetal Acceleration
Since there is a force pulling the object to the center, there must be an acceleration toward the center too. This is called centripetal acceleration (a). The formula is a = v2 / r or a = w2 r.
Calculating Centripetal Force
To find the force (F), we use Newton's second law (F = ma). So, F = mv2 / r or F = mw2 r.
Worked Example:
A stone of mass 0.5 kg is whirled in a horizontal circle of radius 2 m with a linear velocity of 4 m s-1. Calculate the centripetal force.
1. Identify given values: m = 0.5 kg, r = 2 m, v = 4 m s-1.
2. Use the formula: F = mv2 / r.
3. Substitute: F = (0.5 * 42) / 2.
4. Calculate: F = (0.5 * 16) / 2 = 8 / 2 = 4 N.
Key points
- •Angular velocity is measured in radians per second (rad s-1).
- •Linear velocity (v) equals angular velocity (w) multiplied by radius (r).
- •Centripetal force always acts towards the center of the circle.
- •Centripetal acceleration is caused by the change in direction of velocity.
- •Mass, velocity, and radius all affect the strength of the centripetal force.