Uniform Circular Motion

Uniform circular motion refers to the motion of an object traveling in a circular path at a constant speed. While the speed of the object remains constant, its velocity continuously changes because velocity is a vector quantity (it has both magnitude and direction), and the direction changes as the object moves along the circular path.

Key Characteristics of Uniform Circular Motion:

1. Constant Speed:
   The magnitude of the velocity remains the same throughout the motion.
2. Changing Velocity:
   The direction of velocity is always tangential to the circular path. Since the direction changes continuously, the object is accelerating.
3. Centripetal Force:
   A force directed toward the center of the circle acts on the object to keep it moving in the circular path. This force is called the centripetal force.
4. Centripetal Acceleration:
   The acceleration of the object, called centripetal acceleration, points toward the center of the circle and is responsible for changing the direction of the velocity.

Key Equations in Uniform Circular Motion

1. Centripetal Force:
   The force required to keep an object in uniform circular motion is: Fc=mv^2/r  Where:
   • Fc : Centripetal force (N)
   • m: Mass of the object (kg)
   • v: Speed of the object (m/s)
   • r: Radius of the circular path (m)
2. Centripetal Acceleration:
   The acceleration directed toward the center of the circle is: a_c=v²/r   Where:
   • a_c : Centripetal acceleration (m/s²)
   • v: Speed of the object (m/s)
   • r: Radius of the circular path (m)
3. Angular Velocity (ω): The angular velocity of an object is the object’s angular displacement with respect to time, There are three formulas that we can use to find the angular velocity of an object.  
   1. Option 1 :  Angular velocity is expressed as follows: ω= ϴ/t,  Where ω is the angular velocity, θ(theta) is the angular displacement, and t is the change in time t.
   2. Option 2 : In the second method, we recognize that (theta) is given in radians, and the definition of radian measure gives theta = s / r. Also, we can put theta in first angular velocity formula. This will give ω = (s / r) / t , w = s / (rt) , Where s refers to the arc length,  r refers to the radius of the circle
      t  refers to the time taken
   3. Option 3: ω= s / (rt), w = (s / t) (1 / r), we know that  s/t is linear velocity, v = s/t,  Hence, we can rewrite it as w = v (1 / r) = v / r. Hence  ω=v/r  Where ω Angular velocity (rad/s) ,v: Linear speed (m/s) , r: Radius of the circle (m)
4.  Time Period (T):
   The time taken to complete one revolution: T=2πr/v Where:
   • T: Time period (s)
   • r: Radius of the circle (m)
   • v: Speed (m/s)
5. Frequency (f):
   The number of revolutions per second: f=1/T

Examples of Uniform Circular Motion:

1. A satellite orbiting the Earth.
2. The motion of a car around a circular track.
3. A stone tied to a string and swung in a horizontal circle.
4. The blades of a rotating ceiling fan.

Important Points to Note:

• Even though the speed is constant, the object is accelerating because the direction of velocity is changing.
• Centripetal force does not do work because it acts perpendicular to the direction of motion (no displacement in the direction of force).
• If the centripetal force is removed, the object will move tangentially to the circular path due to inertia (Newton's First Law).

Uniform circular motion plays a key role in understanding rotational dynamics and many real-world phenomena involving circular trajectories.