Uniform Circular Motion Lecture Notes

Key Concepts

• Uniform Circular Motion: Object moving in a circle at constant speed.
  • Direction: Changing direction implies change in velocity, hence acceleration.
  • Centripetal Acceleration: Always points towards the center of the circle.
  • Velocity: Speed with a direction; magnitude constant, direction changes.

Formulas to Know

• Centripetal Acceleration ((a_c)):

  • [ a_c = \frac{v^2}{r} ]
  • Doubling velocity results in quadrupling centripetal acceleration.

• Centripetal Force:

  • [ F_c = m \times a_c = \frac{m \times v^2}{r} ]

• Velocity in Circular Motion:

  • [ v = \frac{2 \pi r}{T} ]
  • (T): Period (time for one complete revolution).

• Frequency ((f)):

  • [ f = \frac{1}{T} ]

• Centripetal Acceleration in terms of Radius and Period:

  • [ a_c = \frac{4 \pi^2 r}{T^2} ]

Tension in Circular Motion

• Vertical Circle:

  • At points A, C: ( T = \frac{mv^2}{r} )
  • At point D (bottom): ( T = F_c + mg )
  • At point B (top): ( T = F_c - mg )

• Horizontal Circle with Angle:

  • Tension components: ( T_x ), ( T_y )
  • [ T = \sqrt{T_x^2 + T_y^2} ]
  • ( T_x = \frac{mv^2}{r} )
  • ( T_y = mg )
  • ( \tan \theta = \frac{T_y}{T_x} )

Normal Force on Hill/Valley

• Bottom of Hill (Valley):

  • [ N = F_c + mg ]
  • Normal force maximum here.

• Top of Hill:

  • [ N = mg - F_c ]
  • Normal force minimum here.
  • If ( N < 0 ), object loses contact with ground.

Additional Resources

• Check out example problems and further explanations through links provided in the description of the lecture.
• Videos titled "Normal Force on a Hill" available for practical understanding.