Uniform Circular Motion

Study Notes and Summary

Definition: When an object moves in a circular path with uniform speed, its motion is called uniform circular motion.

Changing Velocity: Even though the speed is constant, the velocity of the object is continuously changing. This is because the direction of motion is continuously changing at every point along the circular path.

Accelerated Motion: Since velocity is changing (due to change in direction), uniform circular motion is an accelerated motion.

Centripetal Acceleration: The acceleration in uniform circular motion is always directed towards the center of the circular path. This is called centripetal acceleration.

Centripetal Force: An unbalanced force is required to produce this centripetal acceleration, which acts towards the center of the circle. This force is called centripetal force. Without this force, the object would move in a straight line (tangential to the circle) due to inertia.

Examples:

  • Motion of the Moon around the Earth.
  • Motion of a satellite in a circular orbit around the Earth.
  • An athlete running on a circular track with constant speed.
  • A stone tied to a string and whirled in a circle.

Velocity in Circular Motion:

For an object moving in a circular path of radius rrr with constant speed v, it completes one revolution (distance = 2Ď€r) in time T (period).
v = \(\frac{2\pi r}{T}\)

Numerical Based Question (if applies)

Scenario: An athlete runs on a circular track of radius 50 m. If they complete one round in 20 seconds, what is their speed?

Solution Strategy (for students):

  • Identify given values: \(r=50 m,T=20 s\) 
  • Distance covered in one round = 2Ď€r
  • Use the formula for speed in circular motion:
    v = \(\frac{2\pi r}{T}\)

Substitute values:
v = \(\frac{2 \times 3.14 \times 50}{20} = \frac{314}{20} = 15.7\, \text{m/s}\)

This MCQ module is based on: Uniform Circular Motion

This assessment will be based on: Uniform Circular Motion

Investigating Centripetal Force and its Dependence on Mass, Velocity, and Radius

Objective: To experimentally verify the relationship between centripetal force, the mass of the object, its speed, and the radius of the circular path.

Materials: Centripetal force apparatus (e.g., a rotating stand with a string, stopper, and hanging weights), stopwatch, meter scale, known masses (for the stopper and hanging).

Procedure:

Setup:
 Attach a stopper (mass mmm) to one end of a string, pass the string through a glass tube, and attach a hanger with known weights (mass M) to the other end. The hanging weights provide the centripetal force:
\(F_c = Mg\)

Varying Velocity (Constant Mass, Constant Radius):

  • Fix the radius r
  • Whirl the stopper in a horizontal circle at a constant speed
  • Measure time for 20–30 revolutions to calculate period T
  • Calculate speed:
    v = \(\frac{2\pi r}{T}\)
  • \(\text{Plot of } F_c \text{ vs. } v^2\)

Varying Mass (Constant Velocity, Constant Radius):

  • Replace the stopper with a different mass ‘m′
    Maintain constant v and r
  • Measure required hanging mass M′
  • \(\text{Plot of } F_c \text{ vs. } m\)

Varying Radius (Constant Mass, Constant Velocity):

  • Change radius r
  • Maintain constant mmm and v
  • Measure required hanging mass
  • \(\text{Plot of } F_c \text{ vs. } \frac{1}{r}\)

Expected Observations:

  • \(F_c \propto v^2 \quad \text{(for constant } m, r\text{)}\)
  • \(F_c \propto m \quad \text{(for constant } v, r\text{)}\)
  • \(F_c \propto \frac{1}{r} \quad \text{(for constant } m, v\text{)}\)

Theoretical Outcomes & Advanced Concepts:

This experiment provides empirical evidence for:
\(F_c = \frac{mv^2}{r}\)

Centripetal vs. Centrifugal:

  • Centripetal force is real and directed towards the center
  • Centrifugal force is fictitious (seen in rotating frames)

Olympiad Connection:
 Real-world and numerical problems involving:

  • Car turning on curved roads
  • Satellites in orbit
  • Tension/friction providing centripetal force
  • Non-inertial frames

2. Observing Uniform Circular Motion and Tangential Velocity

Objective:
 To observe the trajectory of an object when the centripetal force is removed during uniform circular motion.

Materials: Stone, string, open space.

Procedure:

  • Tie a stone to one end of a string
    Whirl the stone in a horizontal circle
  • Release the string at a specific point

Expected Observations:
 The stone moves in a tangential straight line, not inward or outward.

Theoretical Outcomes & Advanced Concepts:

Inertia of Direction (Newton’s First Law):
 Once centripetal force is removed, object continues tangentially due to inertia.

Instantaneous Velocity:
 Direction of tangential motion equals the velocity vector at the release point.

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