Question

Consider a system of three equal-mass particles moving in a plane; their positions are given by r_1= a_1i+ b_1j where a_i and b_i are functions of time with units of position. Particle 1 has a_1= 3t_2+ 7 and b_1=0; particle 2 has a_2= 5t+5 and b_2=4; and particle 3 has a_3=4t andb_3=3t+5. Find the position, velocity, and acceleration of the centre of mass at t=2.5 s. r_cm= v_cm a_cm i+ j m i+ j m/s i+ j m/s^2

Answer 1

A) position vectors

r1 = (3t^2 +7)i + 0j

r2 = (5t +5)i + 4j

r3 = (4t)i + (3t +5) j

centre of mass position vector

Rcm = (m1r1 + m2r2 + m3r3)/(m1 + m2 +m3) = (r1 +r2 +r3 )/3 = ((3t^2 +7)i + (5t +5)i + 4j +(4t)i + (3t +5) j )/3

=> Rcm = [(3t^2 +9t +12)i + (3t + 9)j ]/3 = (t^2 + 3t +4 )i + (t+3)j m

B) velocity vector

Vcm = dRcm/dt = (2t+3)i + 1j m/s

C) acceleration vector

Acm = dVcm/dt = 2i m/s^2