6.4 Period and Frequency for Uniform Circular Motion
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1 6.4 Peiod and Fequency fo Unifom Cicula Motion If the object is constained to move in a cicle and the total tangential foce acting on the total object is zeo, F θ = 0, then (Newton s Second Law), the tangential acceleation is zeo, a θ = 0. (6.4.1) This means that the magnitude of the velocity (the speed) emains constant. This motion is known as unifom cicula motion. The acceleation is then given by only the acceleation adial component vecto a (t) = ω 2 (t) ˆ(t) unifom cicula motion. (6.4.2) Because the speed v = ω is constant, the amount of time that the object takes to complete one cicula obit of adius is also constant. This time inteval, T, is called the peiod. In one peiod the object tavels a distance s = vt equal to the cicumfeence, s = 2π ; thus s = 2π = vt. (6.4.3) The peiod T is then given by 2π 2π 2π T = = =. (6.4.4) v ω ω The fequency f is defined to be the ecipocal of the peiod, 1 ω f = =. (6.4.5) T 2π The SI unit of fequency is the invese second, which is defined as the hetz, s 1 [Hz]. The magnitude of the adial component of the acceleation can be expessed in seveal equivalent foms since both the magnitudes of the velocity and angula velocity ae elated by v = ω. Thus we have seveal altenative foms fo the magnitude of the centipetal acceleation. The fist is that in Equation (6.5.3). The second is in tems of the adius and the angula velocity, a = ω 2. (6.4.6) 1
2 The thid fom expesses the magnitude of the centipetal acceleation in tems of the speed and adius, 2 v a =. (6.4.7) Recall that the magnitude of the angula velocity is elated to the fequency by ω = 2π f, so we have a fouth altenate expession fo the magnitude of the centipetal acceleation in tems of the adius and fequency, a = 4π 2 f 2. (6.4.8) 2
3 A fifth fom commonly encounteed uses the fact that the fequency and peiod ae elated by f = 1/ T = ω / 2π. Thus we have the fouth expession fo the centipetal acceleation in tems of adius and peiod, 4π 2 a =. (6.4.9) T 2 Othe foms, such as 4π 2 2 f / T o 2πω f, while valid, ae uncommon. Often we decide which expession to use based on infomation that descibes the obit. A convenient measue might be the obit s adius. We may also independently know the peiod, o the fequency, o the angula velocity, o the speed. If we know one, we can calculate the othe thee but it is impotant to undestand the meaning of each quantity Geometic Intepetation fo Radial Acceleation fo Unifom Cicula Motion An object taveling in a cicula obit is always acceleating towads the cente. Any adial inwad acceleation is called centipetal acceleation. Recall that the diection of the velocity is always tangent to the cicle. Theefoe the diection of the velocity is constantly changing because the object is moving in a cicle, as can be seen in Figue 6.4. Because the velocity changes diection, the object has a nonzeo acceleation. Figue 6.4 Diection of the velocity fo cicula motion. Figue 6.5 Change in velocity vecto. 3
4 The calculation of the magnitude and diection of the acceleation is vey simila to the calculation fo the magnitude and diection of the velocity fo cicula motion, but the change in velocity vecto, Δv, is moe complicated to visualize. The change in velocity Δv = v(t + Δt) v( t ) is depicted in Figue 6.5. The velocity vectos have been given a common point fo the tails, so that the change in velocity, Δv, can be visualized. The length Δv of the vetical vecto can be calculated in exactly the same way as the displacement Δ. The magnitude of the change in velocity is Δv = 2vsin( Δθ / 2). (6.5.1) We can use the small angle appoximation sin (Δθ / 2 ) Δ θ / 2 to appoximate the magnitude of the change of velocity, The magnitude of the adial acceleation is given by Δv v Δθ. (6.5.2) Δv v Δθ Δθ dθ a = lim = lim = v lim = v = v ω. (6.5.3) Δt 0 Δt Δt 0 Δt Δt 0 Δt dt The diection of the adial acceleation is detemined by the same method as the diection of the velocity; in the limit Δθ 0, Δv v, and so the diection of the acceleation adial component vecto a (t) at time t is pependicula to position vecto v( t ) and diected inwad, in the ˆ -diection. 4
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