Question 21: Oscillations

Question

A point mass is subjected to two simultaneous sinusoidal displacements in x-direction, $${x_1}\left( t \right) = A\sin \omega t$$ and $${x_2}\left( t \right) = A\sin \left( {\omega t + {{2\pi } \over 3}} \right)$$. Adding a third sinusoidal displacement $${x_3}\left( t \right) = B\sin \left( {\omega t + \phi } \right)$$ brings the mass to a complete rest. The values of B and $$\phi $$ are

Accepted Answer

Here, $${x_1} = A\sin \omega t$$
$${x_2} = A\sin \left( {\omega t + {{2\pi } \over 3}} ight)$$
$$	herefore$$ $${x_1} + {x_2} = A\sin \omega t + A\sin \left( {\omega t + {{2\pi } \over 3}} ight)$$
$$ = A\sin \omega t + A\left[ {\sin \omega t\cos {{2\pi } \over 3} + \cos \omega t\sin {{2\pi } \over 3}} ight]$$
$$ = A\sin \omega t + A\left[ {\sin \omega t\left( { - {1 \over 2}} ight) + \cos \omega t\left( {{{\sqrt 3 } \over 2}} ight)} ight]$$
$$ = {A \over 2}\sin \omega t + {{\sqrt 3 } \over 2}A\cos \omega t = A\left[ {\sin \omega t\cos {\pi  \over 3} + \cos \omega t\sin {\pi  \over 3}} ight]$$
$$ = A\sin \left( {\omega t + {\pi  \over 3}} ight)$$
$$\because$$ $${x_1} + {x_2} + {x_3} = 0$$
$$ \Rightarrow {x_3} =  - ({x_1} + {x_2}) =  - A\sin \left( {\omega t + {\pi  \over 3}} ight)$$
$$ = A\sin \left( {\omega t + \pi  + {\pi  \over 3}} ight)$$
$${x_3} = A\sin \left( {\omega t + {{4\pi } \over 3}} ight)$$
$$\because$$ $${x_3} = B\sin (\omega t + \phi )$$
Hence, $$B = A,\,\phi  = {{4\pi } \over 3}$$

Answer

$$\sqrt 2 A,{{3\pi } \over 4}$$

Answer

$$A,{{4\pi } \over 3}$$

Answer

$$\sqrt 3 A,{{5\pi } \over 6}$$

Answer

$$A,{\pi  \over 3}$$

Oscillations

Choose the correct answer: