circular motion final

Circular Motion

  • Definition: Circular motion is the motion of a body following a circular path.

  • Uniform Circular Motion: A specific type of circular motion where a body moves at a constant speed while following a circular path.

    • The body remains equidistant from a fixed central point.

    • Change in direction is due to centripetal force.

      • Centripetal Force: The force acting on a body in circular motion, directed towards the center of the circle.

      • Symbol: FC

Formulas and Concepts in Uniform Circular Motion

  • Causes of forces:

    • Tension (T) in strings can provide the required centripetal force for objects in circular motion.

      • Equation: T = m(v²/r) where:

        • m = mass of the object

        • v = speed of the object

        • r = radius of the circular path

  • Effects of String Breakage: If the string suddenly breaks:

    • Tension disappears, removing the centripetal force.

    • The object will move tangentially to the circle at the moment the string breaks.

Vertical Circular Motion (Rotating String)

  • Forces at different points:

    • At the Top (T + mg = m(v²/r)): The weight of the object (mg) contributes to the tension (T).

    • At the Bottom (T - mg = m(v²/r)): The tension (T) must counteract the weight while providing centripetal force.

  • Variables:

    • m = mass

    • v = speed

    • r = radius of the circle (length of string)

Sample Problems

  1. Model Airplane Scenario:

    • Given: Mass = 0.90 kg, radius = 17 m, speed = 19 m/s

    • Find Tension (T):

      • Formula: T = m(v²/r)

      • Calculation: T = 0.90 kg (19 m/s)² / 17 m

  2. Normie Neutron’s Ball Swing:

    • Given: Length of string = 1.5 m, turns per minute = 120

      • (a) Average velocity:

        • v = circumference / time = (360π / 60)

      • (b) Centripetal Acceleration:

        • a = v² / r = (18.82 m/s) / 1.5 m

  3. Mass in Circular Motion Problem:

    • Given: Mass = 0.100 kg, string length = 75 cm, revolution time = 0.80 s

      • (a) Centripetal acceleration:

        • a = v² / r = (2.95 m/s)² / 0.75 m

      • (b) Tension in String:

        • T = ma = 0.1 kg * a

  4. Vertical Circle Problem:

    • Given: Object mass = 2.2 kg, radius = 1.0 m, time for one revolution = 0.97 s

    • (a) Tension at the top: Apply the forces considering gravity and centripetal acceleration.

    • (b) Tension at the bottom: Similar approach, adjusting for additional gravitational force.