A body is performing SHM, then its:

(a) average total energy per cycle is equal to its maximum kinetic energy.
(b) average kinetic energy per cycle is equal to half of its maximum kinetic energy.
(c) mean velocity for a complete cycle is equal to \(\dfrac{2}{\pi}\) times of its maximum velocity.
(d) root mean square velocity is \(\dfrac{1}{\sqrt{2}}\) times of its maximum velocity.

Choose the correct alternatives:
1. (a), (b), (d)

2. (a), (c)

3. (b), (d)

4. (b), (c), (d)

(1) Hint: Use the formula of kinetic energy and the total energy of SHM.
Let the equation of an SHM is represented as x=asinωt
Assume the mass of the body is m.
(a) Step 1: Find the maximum kinetic energy and average total energy of the particle.
The total mechanical energy of the body at any time t is,
                                        E=12m ω2a2                               ...(i)
Kinetic energy at any instant t is,
                                                                     K=12m v2=12mdxdt2                         v=dxdt
                              =12mω2a2 cos2 ωt                          
                        Kmax=12mω2a2=E                     [ for kmax, cos ωt=1]...(ii)
(b) Step 2: Find the average kinetic energy of the particle.
KE at any instant t is,
                                                K=12mω2a2 cos2 ωt
                      (Kav) for a cycle =12mω2a2cos2ωtav for a cycle
                                              =12mω2a20+12                
                                              =14mω2a2=Kmax2        [from Eq.(ii)]
Step 3: Find the mean velocity and the RMS velocity of the particle.
(c) Velocity =v=dxdt=a ωcos ωt
                                     vmean=vmax+vmin2
                                              =aω+(-aω)2=0                   [For a complete cycle]
                                    vmaxvmean
(d)
vrms=v12+v222=0+a2ω22=aω2
              vrms=vmax2