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Chapter 9: Problem 37

Consider a system of three equal-mass particles moving in a plane; their positions are given by \(a_{i} \hat{\imath}+b_{i} \hat{\jmath},\) where \(a_{i}\) and \(b_{i}\) are functions of time with the units of position. Particle 1 has \(a_{1}=3 t^{2}+5\) and \(b_{1}=0 ;\) particle 2 has \(a_{2}=7 t+2\) and \(b_{2}=2 ;\) particle 3 has \(a_{3}=3 t\) and \(b_{3}=2 t+6 .\) Find the center-of-mass position, velocity, and acceleration of the system as functions of time.

Short Answer

Expert verified
The position of the center of mass is given by \(R_{cm} = (t^{2} + 3t+ \frac{7}{3})\hat{\imath}+ (\frac{2}{3}t+ \frac{8}{3})\hat{\jmath}\), the velocity by \(V_{cm} = (2t+3)\hat{\imath} + (\frac{2}{3})\hat{\jmath}\) and the acceleration by \(a_{cm} = 2\hat{\imath}\)

Step by step solution

01

Compute position of Center of Mass

Given positions of three particles as \(r_1= a_{1}\hat{\imath}+b_{1}\hat{\jmath}\), \(r_2= a_{2}\hat{\imath}+b_{2}\hat{\jmath}\), and \(r_3=a_{3}\hat{\imath}+b_{3}\hat{\jmath}\). The Center of Mass \(R_{cm}\) is calculated as \((a_{cm}, b_{cm})\) where \(a_{cm}= \frac{1}{3}(a_{1} + a_{2}+ a_{3})\) and \(b_{cm} = \frac{1}{3}(b_{1} + b_{2} + b_{3})\). Substituting values, \(a_{cm} = \frac{1}{3}(3t^{2} + 5+ 7t + 2 + 3t) = t^{2} + 3t+ \frac{7}{3}\) and \(b_{cm} = \frac{1}{3}(0 + 2+ 2t + 6) = \frac{2}{3}t+ \frac{8}{3}\). So, \(R_{cm} = (t^{2} + 3t+ \frac{7}{3})\hat{\imath}+ (\frac{2}{3}t+ \frac{8}{3})\hat{\jmath}\)
02

Compute velocity of Center of Mass

Take derivative of \(R_{cm}\) with respect to time, \(V_{cm} = \frac{d}{dt} R_{cm} = \frac{d}{dt} (t^{2} + 3t+ \frac{7}{3})\hat{\imath}+ \frac{d}{dt} (\frac{2}{3}t+ \frac{8}{3})\hat{\jmath} = (2t+3)\hat{\imath} + (\frac{2}{3})\hat{\jmath}\) which is the velocity vector of center of mass
03

Compute acceleration of Center of Mass

Take derivative of \(V_{cm}\) with respect to time, \(a_{cm} = \frac{d}{dt} V_{cm} = \frac{d}{dt} ((2t+3)\hat{\imath} + (\frac{2}{3})\hat{\jmath}) = 2\hat{\imath} which is the acceleration vector of center of mass. Note, the \(\jmath\) component vanishes as the derivative of a constant is zero

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