In order chapter 4 it to be in a straight for the

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Unformatted text preview: ω t ≈ ω t and cos ω t ≈ 1 − (ω t)2 /2. Therefore, v0 cos θ ω t = (v0 cos θ)t. ω x(t) ≈ (96) And, using ω 2 = k/m, y (t) ≈ −(k/m)t2 2 mg k + v0 sin θ 1 ω t = − gt2 + (v0 sin θ)t. ω 2 (97) These are the standard projectile results, as desired. For the above approximations to be valid, we need ω t 1 throughout the entire motion. If we assume that ω (2v0 sin θ/g ) 1, then the above approximations hold for t = 2v0 sin θ/g , in which case we have y ≈ 0 at this time. That is, the projectile has hit the ground and the motion is finished. So “small ω ” means ω g /(v0 sin θ). Now consider large ω . For any t, the x(t) motion is simple harmonic. In order CHAPTER 4. it to be in a straight for the whole motion to be simple harmonic, we need OSCILLATIONS line, 32 so y/x must be a constant. This means that the (mg/k )(cos ω t − 1) term in y (t) must be negligible. We therefore need v0 sin θ ω mg k =⇒ v0 sin θ ω g ω2 g . v0 sin θ =⇒ ω (98) This is what is meant by “large ω .” In this case, both x and y are (essentially) proportional to sin ω t. The projectile reaches a maximum distance from the origin of v0 /ω , and then heads back. The above two conditions on ω can be summed up by saying that the time scale of oscillations without gravity, namely 1/ω , should be much greater than or much less than the time scale of projectile motion without the spring, namely 2v0 sin θ/g . (c) We want y = 0 when x = 0. But x = (v0 cos θ) cos ω t, which is zero when ˙ ˙ t = π /2. The y value at t = π /2 is (mg/k )(0 − 1) + (v0 sin θ/ω )(1). Setting this equal to zero, and using k/m = ω 2 , gives g/ω 2 = v0 sin θ/ω =⇒ ω = g/(v0 sin θ). This is, in a sense, right “between” the two limiting cases above. 4.23. Corrections to the pendulum (a) F = ma in the tangential direction gives mg sin θ = mv dv/dx. Writing dx as dθ , and separating variables and integrating gives θ − So v mg sin θ dθ = θ0 dt = 0 =⇒ v = ± 2g (cos θ − cos θ0 ). (...
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This note was uploaded on 01/19/2014 for the course PHYSICS 251 taught by Professor Brandenberger during the Fall '13 term at McGill.

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