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Circular Motion Under Gravity. Circular Motion under gravity. Loop the loop is possible if the track provides part of the cpforce at the top of the loop ( S T ) The rest of the cpforce is provided by the weight of the rider. Circular Motion Calculations. Centripetal acceleration
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Circular Motion Under Gravity www.assignmentpoint.com
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
Circular Motion Calculations • Centripetal acceleration • Centripetal force www.assignmentpoint.com
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
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
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
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
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
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
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
Producing time traces • 2 ways of producing a voltage analogue of the motion of an oscillating system www.assignmentpoint.com
Time traces www.assignmentpoint.com
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
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
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
SHM4 • Displacement and acceleration are out of phase • a is proportional to - x Hence the minus www.assignmentpoint.com
SHM5 • a = -ω²x equation defines SHM • T = 2π/ω • F = -kx eg a trolley tethered between two springs www.assignmentpoint.com
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