Question

Show that a system of particles moving under the influence of an external force moves as if the force is applied at its centre of mass.

Answer 1

Let us consider r1, r2, r3, .... rn be the position vectors of masses m1, m2, m3 .....mn respectively of n particles system.

Now, according to definition of centre of mass, R = (m1r1 + m2r2 + m3r3 + ..... + mn . rn)/(m1 + m2 + m3 + .... + mn)

Let us consider, M = (m1 + m2 + m3 + ... + mn)

so, R = (m1r1 + m2r2 + m3r3 + .... + mn. rn)/M

or, MR = (m1r1 + m2r2 + m3r3 + .... + mn. rn)

differentiating both sides with respect to time,

M(dR/dt) = m1(dr1/dt) + m2(dr2/dt) + m3(dr3/dt) + ..... mn(drn/dt)

or, MV = m1v1 + m2v2 + m3v3 + ... + mn. vn

again, differentiating both sides with respect to time,

M(dV/dt) = m1(dv1/dt) + m2(dv2/dt) + m3(dv3/dt) + ..... + mn(dvn/dt)

So, Fnet = Ma = m1a1 + m2a2 + m3a3 + .... + mn.an

Where Fnet represents the sum of external force acting on the particles of system.
This equation state that a system of particles moving under the influence of an external force moves as if the force is applied at its centre of mass.

Answer 2

Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic and (c) non periodic motion? Give period for each case of periodic motion (ω is any positive constant) a. sign ωt - cos ωt b. sin² ωt c. 3 Cos (π/4 - ωt) d. cos ωt + cos 3ωt + cos 5ωt e. exp(-ω²t²) f. 1 + ωt + ω²t²