JEE Main · 2025 · 28 Jan 2025 · Shift 2 · Elasticity & SHM

Given below are two statements. One is labelled as
Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Knowing initial position

x_{0}

and initial momentum

p_{0}

is enough to determine the position and momentum at any time t for a simple harmonic motion with a given angular frequency

ω

.
Reason (R) : The amplitude and phase can be expressed in terms of

x_{0}

p_{0}

.
In the light of the above statements, choose the correct answer from the options given below :

(A)Both (A) and (R) are true but (R) is NOT the correct explanation of (A).
(B)(A) is false but (R) is true.
(C)(A) is true but ( $R$ ) is false.
(D)Both (A) and (R) are true and (R) is the correct explanation of (A).✓ Correct

Answer: (D)

Solution

For SHM with angular frequency

ω

, position and momentum are

x = A sin (ω t + ϕ), p = mω A cos (ω t + ϕ)

t = 0

x_{0} = A sin ϕ, p_{0} = mω A cos ϕ

Dividing these gives

tan ϕ = \frac{mω x _{0}}{p _{0}}

sin ϕ = \frac{mω x _{0}}{( mω x _{0} ) ^{2} + p _{0}^{2}}

and from

A = x_{0} / sin ϕ

A = \frac{( mω x _{0} ) ^{2} + p _{0}^{2}}{mω}

Both the amplitude and the phase are thus expressed entirely in terms of

x_{0}

p_{0}

m

and

ω

. Since

m

and

ω

are already known, knowing

x_{0}

and

p_{0}

fully determines

A

and

ϕ

, and hence

x (t)

and

p (t)

at any time. Both the assertion and the reason are true, and the reason correctly explains the assertion.

JEE Main · 2025 · 28 Jan 2025 · Shift 2 · Single correct option