Recording-2025-03-04T16:01:36.840Z

Uniform Circular Motion

• Definition: Uniform circular motion refers to the motion of an object traveling in a circular path at a constant speed.

• Velocity: Although the speed is constant, the velocity is continuously changing due to the constant change in direction.

• Acceleration: In uniform circular motion, the acceleration is always directed toward the center of the circle (centripetal acceleration).

  • The magnitude of centripetal acceleration can be described as:

    • Formula: ( a_c = \frac{v^2}{r} )

Relationship Between Velocity, Radius, and Acceleration

• Radius Effects: The radius of the circular path impacts the motion:

  • Smaller Radius: If the radius is smaller, and the speed remains constant, the object must change direction more quickly, requiring greater centripetal acceleration.

  • Larger Velocity: To maintain circular motion with increased speed, greater acceleration is needed.

Forces in Circular Motion

• Centripetal Force: The force that keeps the object moving in a circular path.

  • Relation: ( F_c = m a_c = m \frac{v^2}{r} )

  • Centripetal force must be applied to maintain circular motion, implying force vectors point towards the center of the circle.

Calculating Centripetal Force and Acceleration

• When calculating centripetal force:

  • Use the mass of the object, velocity, and radius of the circular path.

  • Example Calculation:

    • Given: Mass (m) = 1 kg, Velocity (v) = 2.6 m/s, Radius (r) = 0.75 m.

    • Centripetal acceleration: ( a_c = \frac{(2.6)^2}{0.75} ) results in acceleration and further determines net force (in this case, tension in the string).

Practical Applications of Forces in Circular Motion

• The tension in the string when an object is swung in a circle acts as the centripetal force, maintaining circular motion.

• For instance: If tension = 9 N, it pulls the object forward allowing it to maintain circular trajectory.

  • Mass and radius inversely affect acceleration, and changes in these variables significantly impact motion.

Effects of Changing Variables on Centripetal Acceleration

• Increasing Mass: Increases force proportionally but does not increase acceleration (

  • Force increases proportionally to mass: If mass is doubled, force doubles).

• Increasing Radius: Decreases centripetal acceleration. A larger radius requires less force to maintain circular motion.

  • Example: Doubling the radius halves the centripetal acceleration.

• Changing Velocity: Velocity has a square relationship with centripetal acceleration.

  • If velocity doubles, centripetal acceleration quadruples.

Gravitational Force and Circular Motion

• Centripetal in Celestial Bodies: For planets, gravitational force acts as the centripetal force keeping it in orbit.

  • Formula: For gravitational forces: ( F = \frac{G m_1 m_2}{d^2} )

    • Where ( G ) is the gravitational constant, ( m_1 ) and ( m_2 ) are the two masses, and ( d ) is the distance between their centers.

• Example: Calculation of Earth's orbital path using established parameters without deriving masses explicitly due to known constants.

Summary of Key Concepts

• In summary, while the speed in uniform circular motion stays constant, the velocity changes due to the radial direction change which leads to necessary centripetal acceleration and force adjustments based on radius and mass changes impacting the forces involved in circular motion.