ch05

Chapter 5: Dynamics of Uniform Circular Motion

5.1 Uniform Circular Motion

  • Definition: Movement of an object at a constant speed along a circular path.

  • Key Equation:

    • The relationship between speed (v), radius (r), and period (T) is given by: [ v = \

Example 1: A Tire-Balancing Machine

  • Problem: A car wheel with radius 0.29 m rotates at 830 revolutions per minute.

  • Calculations:

    • First, convert revolutions per minute to seconds: [ \frac{1.2 \times 10^{-3} \text{ min}}{830 \text{ rev/min}} \approx 0.072 \text{ s/rev} ]

    • Use the equation to determine the speed: [ v = \frac{2 \pi (0.29 \text{ m})}{0.072 \text{ s}} \approx 25 \text{ m/s} ]

5.2 Centripetal Acceleration

  • Concept: In uniform circular motion, although speed is constant, the direction of the velocity vector changes, leading to acceleration.

  • Formula:

    • Centripetal acceleration (ac) acts towards the center: [ a_c = \frac{v^2}{r} ]

  • Direction: Centripetal acceleration is directed towards the circle’s center.

Example 2: Which Way Will the Object Go?

  • When an object in uniform circular motion is released, it will move tangentially to the circle (in a straight line), not along the circular path.

Example 3: The Effect of Radius on Centripetal Acceleration

  • Problem: Bobsled track with turns of 33 m and 24 m radius, with a speed of 34 m/s.

  • Calculations:

    • For each radius: [ a_c = \frac{v^2}{r} ] (find values as multiples of g = 9.8 m/s²)

5.3 Centripetal Force

  • Second Law: Newton's Second Law relates net force, mass, and acceleration.

  • Net Force: In uniform circular motion, a net centripetal force is required:

    • Always points to the center of the circle. [ F_c = m a_c = \frac{m v^2}{r} ]

Example 5: The Effect of Speed on Centripetal Force

  • Problem: Determine the tension in a 17 m guideline for a model airplane moving at 19 m/s.

  • Formula Used: [ T = \frac{m v^2}{r} ]

5.4 Banked Curves

  • Frictionless Banked Curve: The normal force provides the centripetal force without friction.

  • Equations:

    • Forces must balance: [ N \cos\theta = mg ] [ N \sin\theta = F_c = \frac{mv^2}{r} ]

Example 8: The Daytona 500

  • Scenario: Cars must travel around a steeply banked turn with a maximum radius of 316 m at a speed where they remain on the track.

  • Calculation: Calculate speed using: [ v = \sqrt{rg \tan\theta} ]

5.5 Satellites in Circular Orbits

  • Orbital Speed: Specific speed is required for a satellite to remain in orbit: [ v = \sqrt{\frac{GM}{r}} ]

Example 9: Hubble Space Telescope

  • Problem: Determine the orbital speed of the Hubble Telescope at 598 km above Earth:

  • Calculation: Utilize gravitational constant and masses to find orbital speed.

5.6 Apparent Weightlessness and Artificial Gravity

  • Concepts: An astronaut experiences weightlessness in free fall due to no normal force acting on them.

  • Artificial Gravity Calculation: For an astronaut to feel their Earth weight, the space station’s rotational speed must match: [ v = \sqrt{rg} ], where r = 1700 m.

5.7 Vertical Circular Motion

  • Forces acting on an object in vertical circular motion change based on the object's position in the circle, affecting the normal force (FN) and gravitational force (mg) relations.