Uniform Circular Motion

When an object moves along a circular path at a constant speed, it’s called uniform circular motion. Even though the speed is constant, the object accelerates because its direction of motion keeps changing!

Key Concepts

  • Centripetal Acceleration: The acceleration is always directed toward the center of the circle. Its magnitude is:
    \( a_c = \frac{v^2}{R} \)   or   \( a_c = \omega^2 R \)

    Here, \( v \) = linear speed, \( \omega \) (omega) = angular speed, and \( R \) = radius of the circle.
  • Angular Speed: How fast the object rotates (measured in radians per second):
    \( \omega = \frac{\Delta \theta}{\Delta t} \)

    Relation to linear speed: \( v = \omega R \).
  • Time Period (T): Time for one full revolution.
    \( T = \frac{2\pi R}{v} = \frac{2\pi}{\omega} \)
  • Frequency (ν): Number of revolutions per second.
    \( \nu = \frac{1}{T} \)

    Connections: \( v = 2\pi R \nu \) and \( \omega = 2\pi \nu \).

Example: Insect in a Circular Groove

Problem: An insect completes 7 revolutions in 100 seconds in a groove of radius 12 cm. Find its angular speed, linear speed, and centripetal acceleration.

Solution:

  • Angular speed: \( \omega = \frac{2\pi \times 7}{100} = 0.44 \, \text{rad/s} \)
  • Linear speed: \( v = \omega R = 0.44 \times 12 = 5.3 \, \text{cm/s} \)
  • Centripetal acceleration: \( a_c = \omega^2 R = (0.44)^2 \times 12 = 2.3 \, \text{cm/s}^2 \)

Note: The acceleration isn’t a constant vector because its direction changes, but its magnitude stays the same!

Important for Exams (NEET)

  1. Centripetal Acceleration Formula: Know \( a_c = \frac{v^2}{R} \) and \( a_c = \omega^2 R \).
  2. Direction of Acceleration: Always points toward the center of the circle.
  3. Angular vs. Linear Speed: Link them with \( v = \omega R \).
  4. Time Period & Frequency: Practice problems connecting \( T \), \( \nu \), \( v \), and \( \omega \).
  5. Real-World Applications: Examples like vehicles turning on roads, planetary orbits, or spinning objects.

Remember: Even if the speed is constant, circular motion involves acceleration. Keep practicing problems to master these concepts!