Uniform Circular Motion Study Notes

Key Concepts

• Uniform Circular Motion: Objects moving in a circle at constant speed.
• Direction of Motion: Can be counterclockwise or clockwise.
• Velocity Vector: Changes direction; at a point, it can be upward or directed west.
• Centripetal Acceleration: Always points towards the center of the circle.
  • Formula: ( a_c = \frac{v^2}{r} )
  • Doubling velocity increases centripetal acceleration by a factor of four.

Velocity and Acceleration

• Uniform Speed: Magnitude of velocity is constant, but direction changes, causing acceleration.
• Acceleration: Change in velocity over time.
• Centripetal Force: Related to centripetal acceleration; keeps the object moving in a circle.
  • Formula: ( F_c = m \cdot \frac{v^2}{r} )

Calculating Velocity and Period

• Velocity Formula: Displacement over time; displacement equals circumference ( 2\pi r ).
• Period (T): Time for one complete revolution; measured in seconds.
• Frequency (f): ( f = \frac{1}{T} )
• Centripetal Acceleration (using period): ( a_c = \frac{4\pi^2 r}{T^2} )

Tension in Circular Motion

• Vertical Circle:
  • Points A, C: Tension ( T \approx \frac{mv^2}{r} )
  • Point D: Tension ( T = F_c + mg )
  • Point B: Tension ( T = F_c - mg )
• Horizontal Circle:
  • Tension ( T = \sqrt{T_x^2 + T_y^2} )
  • ( T_x \approx F_c )
  • ( T_y = mg )
  • Angle ( \theta ): ( \tan \theta = \frac{T_y}{T_x} )

Normal Force on a Hill

• Bottom of Hill:
  • Normal Force ( N = F_c + mg )
  • Maximum normal force due to support and turning.
• Top of Hill:
  • Normal Force ( N = mg - F_c )
  • Minimum normal force; object may lose contact if ( N < 0 ).

Additional Resources

• Check video links for practical examples and problems.
• Suggested videos for further understanding: "Normal Force on a Hill".

Important Formulas

• Centripetal Acceleration: ( a_c = \frac{v^2}{r} )
• Centripetal Force: ( F_c = m \cdot \frac{v^2}{r} )
• Velocity: ( v = \frac{2\pi r}{T} )
• Frequency: ( f = \frac{1}{T} )

These notes are intended to aid in the study of uniform circular motion, covering key concepts, formulas, and problem-solving strategies.