Two simple harmonic motions are represented by y1 = 5[sin 2π t + √3 cos 2π t] and y2 = 5 sin (2π t +(π/4)). The ratio of their amplitudes is

Question

Two simple harmonic motions are represented by y1 = 5[sin 2π t + √3 cos 2π t] and y2 = 5 sin (2π t +(π/4)). The ratio of their amplitudes is (A) 1: 1 (B)  2: 1 (C)  1: 3 (D) √3: 1

Tardigrade Admin · Accepted Answer

Here, y1=5[sin 2π t+√3cos 2π t] =10 [(1/2)sin 2π t+(√3/2)cos 2π t] =10[cos (π/3)sin 2π t+sin(π/3)cos 2π t] =10[sin(2π t+(π/3))] ∴ A1=10 and y2=5 sin(2π t+(π/4)) ∴ A2=5 Hence (A1/A2)=(10/5)=(2/1)

Q. Two simple harmonic motions are represented by $y_{1} = 5 [s in 2 π t + 3 cos 2 π t] an d y_{2} = 5 s in (2 π t + \frac{π}{4})$ . The ratio of their amplitudes is

$1 : 1$

$2 : 1$

$1 : 3$

$3 : 1$

Q. Two simple harmonic motions are represented by y1​=5[sin2πt+3​cos2πt]andy2​=5sin(2πt+4π​). The ratio of their amplitudes is

1:1

2:1

1:3

3​:1

Here, y1​=5[sin2πt+3​cos2πt] =10[21​sin2πt+23​​cos2πt] =10[cos3π​sin2πt+sin3π​cos2πt] =10[sin(2πt+3π​)] ∴A1​=10 and y2​=5sin(2πt+4π​) ∴A2​=5 Hence A2​A1​​=510​=12​

Q. Two simple harmonic motions are represented by $y_{1} = 5 [s in 2 π t + 3 cos 2 π t] an d y_{2} = 5 s in (2 π t + \frac{π}{4})$ . The ratio of their amplitudes is

$1 : 1$

$2 : 1$

$1 : 3$

$3 : 1$