# We could also interpret the result obtained from the

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Unformatted text preview: and 13, one can arrive at the expression (R − r)αθ = − g sin θ ￿ R ￿2 . I 1 + m r2 93 R−r (20) If you compare Equation 14 with equation 20, you will see that the only difference is the factor of (R/(R − r))2 multiplying the geometric factor in the denominator of Equation 20. This is the factor that appears when the curvature of the dish is taken into account. This factor effectively increases the inﬂuence that rolling motion of the ball has on ω . We could also interpret the result obtained from the “planar approximation” in Equation 13 as the limiting case where the radius of the ball is much less than the radius of curvature for the dish, i.e. r ￿ R. In the limit that the dish’s radius is very large, then locally the dish’s surface looks very much like a plane to the ball, and the planar approximation is recovered. Problem 10.7 Once again, we notice that part of our equation for the angular frequency of oscillation depends on the restoring force, and part depends on the ball’s inertia. Which parts of Equation 15 correspond to the restoring force, and which to the ball’s inertia? Problem 10.8 Does ω depend on the ball’s mass? Why or why not? Does it depend on the ball’s radius? Why or why not? If so in either case, what happens to ω when you increase the quantity in question (the mass or radius)? Problem 10.9 If the ball slid within the bowl instead of rolling, how would this change Equation 15? Compare your new Equation 15 to Equation 8 for a simple pendulum. Problem 10.10 As fI increases from zero to one, should ω go up or down? Why? A photograph of the experimental setup for the ball-dish experiment is shown in Figure 8. LoggerPro will record the times at which the ball passes through the photogate and use them to calculate the period of its oscillation. Experiments Experiment 1: Mass on a Spring In our ﬁrst experiment today, we will use the angular frequency of oscillation for a mass on three different springs to determine...
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## This note was uploaded on 09/13/2011 for the course ECONOMICS 101 taught by Professor Gerson during the Spring '11 term at University of Michigan.

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