Question

What will be amplitude and time period of SHM, if equation of displacement is given by - y = a sin2πt + b cos2πt.
1. a
2. b
3. a + b
4. \(\sqrt{a^2+b^2}\)

Answer 1

Correct Answer - Option 4 : \(\sqrt{a^2+b^2}\)

CONCEPT:​

• Time period (T): The time taken to complete one oscillation is called the time period.

The Angular frequency (ω) is given by:

\(ω=\frac{2π}{T}\)

where T is the time period.

• The equation of displacement in SHM is given by:

y = A sin(ωt + ϕ)       .........(i)

where y is the distance from the mean position at any time t, A is amplitude, t is time, and ω is the angular frequency.​

CALCULATION:

Given that y = a sin2πt + b sin2πt

Let a = rcosθ = 5, b = rsinθ

put in the equation

y = rcosθ sin2πt + rsinθ cos2πt = r sin(2πt + θ)

where, a2+ b2 = r2cos2θ + r2sin2θ = r2

or r = √(a2 + b2)

So new equation will be 

y = r sin(2πt + θ) = √(a2 + b2) sin(2πt + θ)

• So amplitude is √(a2 + b2)
• Hence the correct answer is option 4.