The function will represent a periodic motion, if it is identically repeated after a fixed interval of time and will represent simple harmonic motion, if it can be written uniquely in the form of a cos(

2π

T

t+ϕ) or sin(

2π

T

t+ϕ)whereT is the time period.
(a) sin‌ω‌t−cos‌ω‌t
=√2(

1

√2

‌sin‌ω‌t−

1

√2

‌cos‌ω‌t)
=√2(sin‌ω‌t‌cos‌

π

4

−cos‌ω‌t‌sin‌

π

4

)
=√2‌sin(ωt−

π

4

)
∴ It represents simple harmonicwith aperiod T=

2π

ω

and a phase angle (−

π

4

).
(b) sin3ωt=

1

4

(3‌sin‌ω‌t−sin‌3‌ω‌t)
It represents periodic but not simple harmonic motion. Its time period is =

2π

ω

.
(c) 3‌cos(

π

4

−2ωt)=3‌cos(2ωt−

π

4

)
It represents simple harmonic and its time period is

2π

2ω

=

π

ω

(d) cosωt+cos3ωt+cos5ωt
It represents periodic but not simple harmonic motion. Its time period is

2π

ω

. It can be noted that each termrepresents a periodic function with a different angular frequency. Since period is the least interval of time after which a function repeats its value, cos‌ω‌t has a period T=

2π

ω

, cos‌3‌ω‌t has a period

2π

3ω

, cos‌5‌ω‌t has period

2π

5ω

=

T

5

, the last two forms repeat after any integral multiple of their period. Thus each term in the sum repeats itself after T, and hence the sum is a periodic function with a period

2π

ω

(e) exp (–ω2t2) : It is an exponential function which decreases monotonically with increasing time and tends to zero as t→∞ and thus never repeats itself. Therefore it represents non-periodic motion.
(f) 1+ωt+ω2t2 It represents non-periodic motion (physically unacceptable because the function tends to infinity as t→∞).