for simple harmonic motion a (acceleration)  x(displacement)

to be more correct

a=-²x

So a function x=f(t) represent SHM if we can show that

a=double differential of x with respect to time =f''(t) is proportional to x( ie f(t))

 

A function is periodic if i can show that f(t+p)=f(t) for all t. The smallest value of such p is called period of function. [Trignometric functions of (at+b) are always periodic. Ex. sin(5t+3), cos(6t-3) etc. As they themselves are periodic there functions will difinetly be periodic Ex. sin²(5t+3) and e^cos(6t-3) are periodic. ] 

 

All SHM are periodic but a periodic function may or may not represent SHM

 

Take for example

 

(i) f(t)=sin(t)-cos(t)

periodic as f(t+2pi)=f(t) [period 2pi]

f''(t)=-sin(t)+cos(t)=-f(t) Hence SHM

 

(ii)sin³(t)

periodic as f(t+2pi)=f(t)

period 2pi

f''(t)=-3sin³(t)+6sin(t)cos²(t)

     =6sin(t)-9sin³(t)

You can't write it in form of -k*sin³(t) hence does not represent

 

(iii)This one is not clear

 

(iv) f(t) = cos(t)+cos(3t)+cos(5t)

cos(t) has a period of 2pi

cos(3t) has a period of 2pi/3

 (hence it will repeat after every 2pi)

cos(5t) has a period 2pi/5

 (hence it will repeat after every 2pi)

Hence the period of this function is 2pi

f''(t)=-cos(t)-9cos(3t)-25cos(5t)

You can't write it in form of -k*f(t) hence does not represent SHM

 

(v) and (vi) they are not clear