Uniform Circular Motion Lecture Notes

Key Concepts

• Uniform Circular Motion: Motion of objects in a circular path at a constant speed.
• Centripetal Acceleration: Always directed towards the center of the circle.
  • Formula: ( a_c = \frac{v^2}{r} )
  • When velocity ((v)) doubles, centripetal acceleration increases by a factor of four.
• Acceleration: Occurs due to change in direction of velocity, even if speed is constant.

Newton's Laws and Circular Motion

• Newton's Second Law: Net force equals mass times acceleration.
• Centripetal Force: ( F_c = m \cdot a_c = \frac{m v^2}{r} )
  • This is the force needed to keep an object moving in a circle.

Calculating Velocity and Acceleration

• Velocity (v): Displacement over time; for circular motion, it's the circumference over period.
  • Circumference: ( 2\pi \cdot r )
  • Period ((T)): Time for one full revolution.
  • Frequency ((f)): ( f = \frac{1}{T} ) (measured in Hertz)
  • Alternative formula for acceleration: ( a_c = \frac{4\pi^2 r}{T^2} )

Tension in Circular Motion

• Vertical Circle:
  • At points A and C: Tension force (\approx \frac{m v^2}{r})
  • At point D (bottom): Tension = centripetal force + weight force.
  • At point B (top): Tension = centripetal force - weight force.
• Horizontal Circle:
  • Tension force can have both X and Y components.
  • ( T = \sqrt{T_x^2 + T_y^2} )
  • ( T_y = mg )
  • ( \tan(\theta) = \frac{T_y}{T_x} )

Special Cases

• Object on a Hill or Valley:
  • Bottom of Valley:
    • Normal Force = centripetal force + weight force.
  • Top of Hill:
    • Normal Force = weight force - centripetal force.
  • If velocity is too high, object loses contact with the ground.

Practice

• Check description for links to example problems and videos related to these formulas.
• Specific video recommendations:
  • "Normal Force on a Hill" on YouTube for more detailed examples.