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Chapter 9: Problem 15

Three equal masses lie at the corners of an equilateral triangle of side \(L\). Find the center of mass.

Short Answer

Expert verified
The center of mass of the three equal masses at the corners of the equilateral triangle lies at the midpoint of the line drawn from any corner to the midpoint of the opposite side.

Step by step solution

01

Identify pair of masses

First, consider any pair of masses. Since all three have equal magnitudes and are symmetrically placed, the selection doesn't affect the final outcome.
02

Find center of mass for a pair

Since the two masses being considered are equal, their center of mass lies in the middle of the line connecting them. In the case of an equilateral triangle, this line is of length \(L\), so the center of mass is \(L/2\) away from each mass.
03

Extend the consideration to the third mass

Now, consider the center of mass of these two masses and the third mass as a separate system. Since the third mass is equal to the combined masses of the other two, the center of mass of the system lies \(L/2\) away from the third mass, that is, at the midpoint of the line connecting the center of the two masses and the third mass.

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Most popular questions from this chapter

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 \(a=0 ?\)

High-speed photos of a 220 - \(\mu \mathrm{g}\) flea jumping vertically show that the jump lasts \(1.2 \mathrm{ms}\) and involves an average vertical acceleration of \(100 g .\) What (a) average force and (b) impulse does the ground exert on the flea during its jump? (c) What's the change in the flea's momentum during its jump?

A proton moving at \(6.9 \mathrm{Mm} / \mathrm{s}\) collides elastically head-on with a second proton moving in the opposite direction at \(11 \mathrm{Mm} / \mathrm{s}\). Find their subsequent velocities.

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Masses \(m\) and \(3 m\) approach at the same speed \(v\) and collide head-on. Show that mass \(3 m\) stops, while mass \(m\) rebounds at speed \(2 v\).
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