TS EAMCET · Physics · Capacitance

The amplitude of the wave resulting from the superposition of three waves given by $x_{1} = A \cos ω t, x_{2} = 2 A \sin ω t$ and $x_{3} = \sqrt{2} A \cos (ω t + \frac{π}{4})$ is

A $\sqrt{7} A$
B $\sqrt{5} A$
C $(3 + \sqrt{2}) A$
D $\sqrt{2} A$

Verified Solution

Answer & Solution

Correct Answer

(B) $\sqrt{5} A$

Step-by-step Solution

Detailed explanation

The resultant displacement can be calculated by, x=x1+x2+x3, ⇒x=Acosωt+Asinωt+2Acosωt+π4⇒x=Acosωt+2Asinωt+2Acosωtcosπ4-2Asinωtsinπ4⇒x=Acosωt+2Asinωt+Acosωt-Asinωt⇒x=2Acosωt+Asin`…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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List - I			List - II
A	Gravitational potential	I	$M^{0} L^{2} T^{- 2} K^{- 1}$
B	Stefan's constant	II	$M^{0} L^{2} T^{- 2}$
C	Permittivity	III	${ML}^{0} T^{- 3} K^{- 4}$
D	Specific heat capacity	IV	$M^{- 1} L^{- 3} T^{4} I^{2}$

The amplitude of the wave resulting from the superposition of three waves given by x1=Acosωt, x2=2Asinωt and x3=2Acosωt+π4 is

Step-by-step Solution

The amplitude of the wave resulting from the superposition of three waves given by $x_{1} = A \cos ω t, x_{2} = 2 A \sin ω t$ and $x_{3} = \sqrt{2} A \cos (ω t + \frac{π}{4})$ is