Notes on Circular Motion and Speed Calculations

Uniform Circular Motion

• Definition: Motion around at a constant speed in a circle.
• Radius of orbit: 1.5 nanometers

Angular Displacement

• Angular displacement (Theta): Angle turned through, usually measured in radians.
• Given: 40 degrees
  • Convert to radians:
    • Formula:
      [ \text{radians} = \frac{40}{360} \times 2\pi \approx 0.70 \text{ radians} ]

Distance Moved by Electron

• Arc Length (s): Distance traveled along the circular path.
• Formula:
  [ s = R \times \Theta ]
• Where:
  • s = Arc length (in meters)
  • R = Radius (in meters)
  • Theta = Angular displacement (in radians)
• Calculation:
  [ s = (1.5 \times 10^{-9} \text{ m}) \times 0.70 \approx 1.05 \times 10^{-9} \text{ m} ]

Circular Motion Around Earth

• Time period (T): Time for one complete orbit.
• Radius of orbit: Given in kilometers.
• Circumference of circular path:
  [ C = 2\pi R ]
• Speed of satellite:
  [ v = \frac{C}{T} = \frac{2\pi R}{T} ]
• Given time period: 24 hours
  • Convert to seconds:
    [ 24 \text{ hours} = 24 \times 60 \times 60 = 86400 ext{ seconds} ]
• Calculation for speed:
  [ v = \frac{2\pi \times R}{86400} \approx 3770 \text{ m/s} ]

Gear in Car

• Rotations per second: 300 revolutions/second.
• This value represents frequency (f): 300 Hertz.
• Diameter of disk: Given.
• To calculate speed at the edge of the disk:
  • Formula:
    [ v = \frac{C}{T} ]
• Given:
  [ T = \frac{1}{300} ext{ seconds} ]
• Diameter conversion:
  • Convert diameter from centimeters to meters, and find radius:
    [ R = \frac{ ext{Diameter}}{2} ]
• Final calculation of speed:
  [ v = 2\pi R \times \text{Frequency} \approx 113 \text{ m/s} ]

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Summary

• Important equations:
  • Arc length: [ s = R \times \Theta ]
  • Speed: [ v = \frac{2\pi R}{T} ] or [ v = 2\pi R \times f ]

Key Concepts

• Convert angular displacement to radians for calculations.
• Understand the relationship between distance, speed, and time in circular motion.