Question

The displacement of a particle is represented by the
equation y = sin³ ⍵t. The motion is
(a) non-periodic
(b) periodic but not simple harmonic
(c) simple harmonic with period 2π/⍵
(d) simple harmonic with period π/⍵

Answer 1

Answer:

(b) periodic but not shm

Explanation:

Given the equation of displacement of the particle, y=sin

3

ωt

We know sin3θ=3sinθ−4sin

3

θ

Hence, y=

4

(3sinωt−4sin3ωt)

⇒4

dt

dy

=3ωcosωt−4×[3ωcos3ωt]

⇒4×

dt

2

d

2

y

=−3ω

2

sinωt+12ωsin3ωt

⇒

dt

2

d

2

y

=

4

−3ω

2

sinωt+12ωsin3ωt

⇒

dt

2

d

2

y

is not proportional to y.

Hence, the motion is not SHM.

As the expression is involving sine function, hence it will be periodic.

Also sin

3

ωt=(sinωt)

3

=[sin(ωt+2π)]

3

=[sin(ωt+2π/ω)]

3

Hence, y=sin

3

ωt represents a periodic motion with period 2π/ω.

Answer 2

Answer:

I am in class 5 so I don't know about it