Question

Figure 9-24 shows an overhead view of four particles of equal mass sliding over a frictionless surface at constant velocity. The directions of the velocities are indicated; their magnitudes are equal. Consider pairing the particles. Which pairs form a system with a center of mass that (a) is stationary, (b) is stationary and at the origin, and (c) passes through the origin?

Short answer

a) Pairs from the system with a center of mass that is stationary are ac,bc and dc.

b) Pairs from the system with a center of mass that is stationary and is at the origin are bc

c) Pairs from the system with a center of mass that passes through the origin are ad and bd

Step-by-step solution

The given data

Coordinates of the four particles are:

$$\begin{gathered} \mathrm{a}=\left(-4\mathrm{m},2\mathrm{m}\right) \\ \mathrm{b}=\left(4\mathrm{m},2\mathrm{m}\right) \\ \mathrm{c}=\left(-4\mathrm{m},-2\mathrm{m}\right) \\ \mathrm{d}=\left(4\mathrm{m},-2\mathrm{m}\right) \end{gathered}$$

Understanding the concept of the center of mass

The center of mass of two particles of equal masses lies exactly at the center of the tile joining them. If two particles are moving in the opposite direction with equal speeds, then their center of mass will be stationary. From the diagram, we have taken the pair of two masses and compared their velocity and determined the position of their center of mass.

a) Calculation of the system pairs when the center of mass is stationary

From the figure, we can see that particles a and c have the same speed, but their direction is opposite to each other. From this, we can say that pairs a and c have a stationary center of mass.

From the figure, we can see that particles b and c have the same speed, but their direction is opposite to each other. From this, we can say that the pairs b and c have a stationary center of mass.

From the figure,we can seethat particlesd and c have the same speed, but their direction is opposite to each other. From this, we can say that the pairs d and c have the center of mass to be stationary.

Hence, the pairs having a stationary center of mass are ac,bc and dc.

b) Calculation of the system pairs when the center of mass is stationary and at the origin

If we draw a line joining particles b and c, that line will pass through the origin. As both are at an equal distance from the origin, the line joining particles b and c will have an origin as their mid-point.

When two particles are moving in opposite directions with equal speeds, their center of mass will be stationary.

The pair having the stationary center of masses are situated at the origin is bc

c) Calculation of the system pairs when the center of mass passes through the origin

Consider pair of particles a and d, if we draw a line joining particles a and d, that line will pass through the origin. As both are at an equal distance from the origin, the line joining particles a and d will have the origin as their mid-point.

As they move with some speed in the given direction their center of mass will pass through the origin.

Consider the pair of particles b and d, if we draw a line joining particles b and d, it will be parallel to the y- axis. As both are equidistant from the x-axis, their mid-point will lie on the x-axis.

As they move with some speed in the given direction their center of mass will pass through the origin.

The pair having the center of mass passing through the origin is ad and bd