Chapter 13: Problem 7
Why is critical damping desirable in a car's suspension?
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Chapter 13: Problem 7
Why is critical damping desirable in a car's suspension?
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A particle of mass \(m\) has potential energy given by \(U=a x^{2}\), where \(a\) is a constant and \(x\) is the particle's position. Find an expression for the frequency of simple harmonic oscillations this particle undergoes.
an ornithologist who knows you've studied physics. She asks you for a noninvasive way to measure birds" masses. You propose using a bird feeder in the shape of a 50 -cm-diameter disk of mass \(340 \mathrm{~g}\), suspended by a wire with torsional constant \(5.00 \mathrm{~N} \cdot \mathrm{m} / \mathrm{rad}(\mathrm{Fig} .13 .35)\). Two birds land on opposite sides and the feeder goes into torsional oscillation at \(2.6 \mathrm{~Hz}\). Assuming the birds have the same mass, what is it?
This problem explores the nonlinear pendulum discussed P qualitatively in Conceptual Example 13.1. You can tackle this problem if you have experience with your calculator's differential-equation solving capabilities or if you've used a software program like Mathematica or Maple that can solve differential equations numerically. On page 252 we wrote Newton's law for a pendulum as \(1 d^{2} \theta / d t^{2}=-m g L \sin \theta\). (a) Rewrite this equation in a form suitable for a simple pendulum, but without making the approximation \(\sin \theta \cong \theta\). Although it won't affect the form of the equation, assume that your pendulum uses a massless rigid rod rather than a string, so it can turn completely upside down without collapsing. (b) Enter your equation into your calculator or software, and produce graphical solutions to the equation for the situation where you specify the initial kinetic energy \(K_{0}\) when the pendulum is at its bottommost position. In particular, describe solutions for (i) \(K_{0}\) \(K_{0} \leqslant U_{\max }\), and (iii) \(K_{0}>U_{\max } .\) Here \(U_{\max }\) is the maximum pos- (ii) \(K_{0} \leqslant U_{\max }\), and (iii) \(K_{0}>U_{\max } .\) Here \(U_{\max }\) is the maximum pos- sible potential energy for the system, which occurs when the pendulum is completely upside down; \(U_{0}=2 L m g\), where \(L\) is the pendulum's length.
A \(1600-\mathrm{kg}\) car is going at \(90 \mathrm{~km} / \mathrm{h}\). It's got bad shock absorbers, and it's executing vertical SHM with amplitude and frequency of approximately \(15 \mathrm{~cm}\) and \(1.1 \mathrm{~Hz}\), respectively. Estimate the percentage of the car's kinetic energy that is in the oscillation.
Derive the period of a simple pendulum by considering the horizontal displacement \(x\) and the force acting on the bob, rather than the angular displacement and torque.
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