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Chapter 9: Problem 14
A 28 -kg child sits at one end of a 3.5 -m-long seesaw. Where should her \(65-\mathrm{kg}\) father sit so the center of mass will be at the center of the seesaw?
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A thin rod extends from \(x=0\) to \(x=L\). It carries a nonuniform mass per unit length \(\mu=M x^{a} / L^{1+a},\) where \(M\) is a constant with units of mass, and \(a\) is a non-negative dimensionless constant. Find expressions for (a) the rod's mass and (b) the location of its center of mass. (c) Are your results what you expect when \(a=0 ?\)
Model rocket motors are specified by giving the impulse they "provide, in \(\mathrm{N} \cdot \mathrm{s}\), over the entire time the rocket is firing. The table below shows the results of rocket-motor tests with different motors used to launch rockets of different masses. Determine two data-based quantities that, when plotted against each other, should give a straight line and whose slope should allow you to determine \(g .\) Plot the data, establish a best-fit line, and determine \(g\). Assume that the maximum height is much greater than the distance over which the rocket motor is firing, so you can neglect the latter. You're also neglecting air resistance- -but explain how that affects your experimentally determined value for \(g\). $$\begin{array}{|l|r|r|r|r|r|} \hline \text { Impulse, } J(\mathrm{N} \cdot \mathrm{s}) & 4.5 & 7.8 & 4.5 & 7.8 & 11 \\ \hline \begin{array}{l}\text { Rocket mass }(\mathrm{g}) \\ \text { (including motor) }\end{array} & 180 & 485 & 234 & 234 & 485 \\ \hline \begin{array}{l}\text { Maximum height } \\\\\text { achieved (m) }\end{array} & 22 & 13 & 19 & 51 & 23 \\\\\hline\end{array}$$
You're an accident investigator at a scene where a drunk driver in a \(1600-\mathrm{kg}\) car has plowed into a \(1300-\mathrm{kg}\) parked car with its brake set. You measure skid marks showing that the combined wreckage moved 25 m before stopping, and you determine a frictional coefficient of \(0.77 .\) What do you report for the drunk driver's speed just before the collision?
In a railroad switchyard, a 56 -ton freight car is sent at \(7.0 \mathrm{mi} / \mathrm{h}\) toward a 31 -ton car moving in the same direction at \(2.6 \mathrm{mi} / \mathrm{h}\). (a) What's the speed of the cars after they couple? (b) What fraction of the initial kinetic energy was lost in the collision?
A cylindrical concrete silo is \(4.0 \mathrm{m}\) in diameter and \(30 \mathrm{m}\) high. It consists of a \(6000-\mathrm{kg}\) concrete base and \(38,000-\mathrm{kg}\) cylindrical concrete walls. Locate the center of mass of the silo (a) when it's empty and (b) when it's two-thirds full of silage whose density is \(800 \mathrm{kg} / \mathrm{m}^{3} .\) Neglect the thickness of the walls and base.
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