Circular Motion Revision - A Level Physics

Radian Measure of an Angle

• To convert degrees to radians: multiply by ( \pi \div 180^{\circ} ).
• To convert radians to degrees: multiply by ( 180^{\circ} \div \pi ).
• Key conversions:
  • 360° = 2π radians
  • 180° = π radians
  • 90° = π/2 radians

Time Period

• Time period (T): Time to complete one full orbit.
• Frequency (f) = (\frac{1}{T}).
• Velocity (v) = (\frac{2\pi r}{T}), where r is the radius.

Angular Velocity (ω)

• Angular velocity (ω) indicates angular displacement per unit time.
• ω = (\frac{2\pi}{T}) or ω = 2πf
• Linear velocity and angular velocity relation: (v = ωr).

Unit Conversion

• RPM (revolutions per minute) to radians per second: multiply by 2π, divide by 60.
• Example: 200 RPM = 21 radians/second.

Centripetal Force

• Centripetal force: net force causing circular motion, directed toward the center, perpendicular to linear velocity.
• Examples: tension, gravitational force, frictional force.
• Formula: (F = \frac{mv^2}{r})
  • Centripetal acceleration: (a = \frac{v^2}{r})

Acceleration in Circular Motion

• Linear velocity changes direction, not magnitude.
• Centripetal acceleration changes direction of motion.
• Work done: zero because force is perpendicular to displacement.
• (F = mω^2r) and (a = ω^2r)

Experiment to Investigate Circular Motion

• Setup: mass on a string, spun through a cylinder to create circular motion.
• Measure mass (m), radius (r), and time 10 oscillations to reduce uncertainty.
• Calculate T, v, and (v^2).
• Plot graph of force (mg) vs (v^2).
  • Straight line through origin confirms formula.
  • Slope gives mass divided by radius.

Circular Motion at an Angle

• Example: car turning. Normal reaction (R) and weight (mg).
• Components: (R\cos\theta = mg), (R\sin\theta = \frac{mv^2}{r})
• Conical pendulum: similar analysis.

Solving Problems involving Angled Circular Motion

• Rearrange components to find unknowns.
• Example: (R\cos\theta = mg), find R and analyze further.
• Result: (v = \sqrt{gr\tan\theta})

Vertical Circular Motion

• Example: washing machine drum.
• Positions differ based on direction of forces.
• Position 1: Forces same direction (F = mv^2/r = mg + R)
  • (R_{1} = mv^2/r - mg)
• Position 2: Forces opposite direction (F = mv^2/r = R - mg)
  • (R_{2} = mv^2/r + mg)
• R greater at position 2.