Rotation and Torque

Rotation and Torque - PHY121 Physics for the Life Sciences...

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Lecture 7 1 PHY121 – Physics for the Life Sciences I Lecture 7 1. Rotational Dynamics : Torque 2. Newton’s Law for Rotations 3. Rotational Inertia: Moment of Inertia Note: set your Clicker to Channel 21 2/14/2012
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Lecture 7 2 Recap: Newton’s Gravity Gravitational attraction is the force acting between objects that have a “ gravitational charge ”, i.e. gravitational mass (one of the mysterious facts of nature is that inertial mass and gravitation charge are proportional (or identical – in SI units) to another! This, among other facts, led Einstein to the Theory of Relativity) – Force of Gravity between point-masses m 1 and m 2 at distance r 12 from one another: Magnitude : Gm 1 m 2 / r 12 2 , INIFINTE RANGE with G = 6.673×10 –11 Nm 2 /kg 2 • Direction: attractive; notation: r 1 2 is pointing outwards •F o r SPHERICALLY SYMMETRIC objects: as if all mass was at center! – e.g. on mass m by M E of Earth at sealevel:   1 ,1 on 2 1 12 2 2 2 G mm r G  F r  2 E G E R mM G F j extremely weak for “normal” masses 2 E E GM g R  11 24 2 6 6.67 10 5.97 10 6.38 10  2 9.80m/s m g j 2/14/2012
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Orbital Motion around the Earth: Satellites Satellite motion: motion of a mass m around another massive body of mass M – If only Earth gravity exits, and the orbit is circular (with radius r ): • Thus, the orbital speed is constant: uniform circular motion! – for a circular orbit v = 2 π r/T and: – This is a special case of Kepler’s Period Law (Kepler’s 3 rd Law): • The square of the orbital periods T 2 are proportional to the 3 rd power of the orbital semi-major axes a : 2/14/2012 Lecture 7 3 2 G F M G m r 2 23 4 Ta GM Net F 2 r m v M vr G  GM v r 2 r T 2 4 Tr GM Note: higher orbits less speed ! a m centripetal uniform circular motion 
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Comparing the orbital speeds of two satellites in different circular Earth orbits: the higher satellite … 2/14/2012 Lecture 7 4 A. B. C. 14% 9% 78% A. is faster than the satellite in the lower orbit B. is slower than the satellite in the lower orbit C. one cannot say: the speed does not depend on the height, but simply on the speed of the rocket that brought the satellite into orbit …
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Lecture 7 5 Example: Angular Kinematics a CD player is spinning up from zero to a final (angular) speed of 7200 rpm , with a constant angular acceleration of 190 rad/s 2 . Q1: How long did it take to spin up to its final speed? – We know the final speed ( ω f =7200 rpm ), the initial speed ( ω 0 =0 ) and the acceleration ( α = 190 rad/s 2 ): Q2: How many turns has it made in the first 10.0 s ? – Two distinct periods: – the spin-up period, lasting 3.97 s