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Circular Motion Under Gravity: Calculations, Angles, and Forces

Explore circular motion concepts under gravity including centripetal acceleration, force, and angles in radians. Learn about angular speed, force, and acceleration in circular motion. Discover the conical pendulum and periodic motion principles.

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Circular Motion Under Gravity: Calculations, Angles, and Forces

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  1. Circular Motion Under Gravity www.assignmentpoint.com

  2. Circular Motion under gravity • Loop the loop is possible if the track provides part of the cpforceat the top of the loop ( ST ) • The rest of the cpforce is provided by the weight of the rider www.assignmentpoint.com

  3. Circular Motion Calculations • Centripetal acceleration • Centripetal force www.assignmentpoint.com

  4. Angles in circular motion • Radians are units of angle • An angle in radians = arc length / radius • 1 radian is just over 57º • There are2π = 6.28 radians in a whole circle www.assignmentpoint.com

  5. Angular speed • Angular speed ω is the angle turned through per second • ω = θ/t = 2π / T • 2π = whole circle angle • T = time to complete one revolution T = 2π/ω = 1/f f = ω/2π www.assignmentpoint.com

  6. Force and Acceleration • v = 2π r / T and T = 2π / ω • v = r ω • a = v² / r = centripetal acceleration • a = (r ω)² / r = r ω² is the alternative equation for centripetal acceleration • F = m r ω² is centripetal force www.assignmentpoint.com

  7. Weightlessness • True lack of weight can only occur at huge distances from any other mass • Apparent weightlessness occurs during freefall where all parts of you body are accelerating at the same rate www.assignmentpoint.com

  8. Weightlessness These astronauts are in freefall Red Arrows pilots experience up to 9g (90m/s²) This rollercoaster produces accelerations up to 4g (40m/s²) www.assignmentpoint.com

  9. The conical pendulum • The vertical component of the tension (Tcosθ) supports the weight (mg) • The horizontal component of tension (Tsinθ) provides the centripetal force www.assignmentpoint.com

  10. Periodic Motion • Regular vibrations or oscillations repeat the same movement on either side of the equilibrium position f times per second (f is the frequency) • Displacement is the distance from the equilibrium position • Amplitude is the maximum displacement • Period (T) is the time for one cycle or or 1 complete oscillation www.assignmentpoint.com

  11. Producing time traces • 2 ways of producing a voltage analogue of the motion of an oscillating system www.assignmentpoint.com

  12. Time traces www.assignmentpoint.com

  13. Simple Harmonic Motion1 • Period is independent of amplitude • Same time for a large swing and a small swing • For a pendulum this only works for angles of deflection up to about 20º www.assignmentpoint.com

  14. SHM2 • Gradient of displacement v. time graph gives a velocity v. time graph • Max veloc at x = 0 • Zero veloc at x = max www.assignmentpoint.com

  15. SHM3 • Acceleration v. time graph is produced from the gradient of a velocity v. time graph • Max a at V = zero • Zero a at v = max www.assignmentpoint.com

  16. SHM4 • Displacement and acceleration are out of phase • a is proportional to - x Hence the minus www.assignmentpoint.com

  17. SHM5 • a = -ω²x equation defines SHM • T = 2π/ω • F = -kx eg a trolley tethered between two springs www.assignmentpoint.com

  18. Circular Motion and SHM T = 2π/ω • The peg following a circular path casts a shadow which follows SHM • This gives a mathematical connection between the period T and the angular velocity of the rotating peg www.assignmentpoint.com

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