Question

In a free vibration the acceleration vector leads the displacement vector by

Answer 1

In a free vibration the acceleration vector leads the displacement vector by π radians or 180°

Explanation:

In free vibration, we know that the equation of displacement at time t is given by,

x(t) = X sin (wt + Ф)   ............................eq1

which is the equation of displacement vector

where X =  initial displacement

w = natural frequency

Ф = phase different

differentiating this equation w.r.t t we get

x'(t) = Xwcos(wt + Ф)

which is the velocity vector.

again differentiating w.r.t. t we get

x"(t) = - Xw²sin(wt + Ф)

=> x"(t) = Xw²sin( wt + Ф - π)   ....................eq2

which is the equation of acceleration vector.

comparing equation 1 and 2 we get that the acceleration vector leads the displacement vector by π radians or 180°.