JEE Main 2025 Apr 7 Shift 1, Physics Q1: Beats

JEE Main2025Apr 7, Shift 1Physics

Two harmonic waves moving in the same direction superimpose to form a wave $x = a cos (1.5 t) cos (50.5 t)$ where $t$ is in seconds. Find the period with which they beat (close to nearest integer)

A
$6$ s
B
$4$ s
C
$1$ s
D
$2$ s

Solution

Rewrite the product as a sum using the identity $cos A cos B = \frac{1}{2} [cos (A - B) + cos (A + B)]$ :

$x = \frac{a}{2} cos (52 t) + \frac{a}{2} cos (49 t)$

Identify the two constituent frequencies from $2 π f = ω$ :

$f_{1} = \frac{52}{2 π} Hz, f_{2} = \frac{49}{2 π} Hz$

Beat frequency is the absolute difference:

$f_{beat} = f_{1} - f_{2} = \frac{3}{2 π} Hz$

Beat period:

$T_{beat} = \frac{1}{f _{beat}} = \frac{2 π}{3} \approx 2.09 s$

Rounded to the nearest integer: $T \approx 2$ s.

$∴$ Option (4).

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JEE Main 2025 Apr 7 Shift 1, Physics Q1: Beats

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