Question

Time period of a particle in SHM depends on the force constant k and mass m of the particle: T = 2π √ (m/k) A simple pendulum executes SHM approximately. Why then is the time period of a pendulum independent of the mass of the pendulum?

Answer 1

Answer:

Explanation:

Time period of a particle doesnot depend on force constant , K is called SHM constant , in case of a spring executing SHM it is spring constant , while it can be anything else for other kind of particles . SIMPLE PENDULUM executes SHM but its motion isnot linear like that of spring ( which executes linear motion ) . simple pendulum executes rotational motion , so rotational variables are reqd. here ,

                   Assume that the pendulum has covered an angle Y then

 the torque acting on the pendulum is mglsinY(L BE THE LENGTH OF THE STRING) as Y is very small we can assume the torque to be mglY ,

now this torque acts as the restoring force for the pednulum  so its direction will always be oposite to the motion of the pendulum , hence its -mglY ( negative sign indicates opposite direction of the torque to the motion of the pendulum )

    now form rotational mechanics we know , TORQUE = MOMENT OF INERTIA×ANGULAR ACC. (α)

 in this case m.i. = mL²

   HENCE , α= -mglY ÷ mL²

  i.e. α = -g/l    again from shm we know α = -ω²×l

   thus we get ω² = g/l

   and from here time period is = 2π÷ω    = 2π×√(l/g)