A point mass is subjected to two simultaneous sinusoidal displacements in -direction, and Adding a third sinusoidal displacement brings the mass to a complete rest. The values of B and are [2011]

Question

A point mass is subjected to two simultaneous sinusoidal displacements in $x$ -direction, $x_{1} (t) = A \sin ω t$ and $x_{2} (t) = A \sin (ω t + \frac{2 π}{3}) .$ Adding a third sinusoidal displacement $x_{3} (t) = B \sin (ω t + ϕ)$ brings the mass to a complete rest. The values of B and $ϕ$ are [2011]

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(2)

Two sinusoidal displacements $x_{1} (t) = A \sin ω t$ and $x_{2} (t) = A \sin (ω t + \frac{2 π}{3}) .$ have amplitude $A$ each, with a phase difference of $\frac{2 π}{3}$ . It is given that sinusoidal displacement $x_{3} (t) = B (\sin ω t + ϕ)$ brings the mass to a complete rest. This is possible when the amplitude of third $B = A$ and is having a phase difference of $ϕ = \frac{4 π}{3}$ with respect to $x_{1} (t)$ as shown in the figure.

Solution

1 $\sqrt{2} A, \frac{3 π}{4}$

2 $A, \frac{4 π}{3}$

3 $\sqrt{3} A, \frac{5 π}{6}$

4 $A, \frac{π}{3}$

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Solution

Q. A point mass is subjected to two simultaneous sinusoidal displacements in x-direction, x1(t)=Asinωt and x2(t)=Asin(ωt+2π3). Adding a third sinusoidal displacement x3(t)=Bsin(ωt+ϕ) brings the mass to a complete rest. The values of B and ϕ are [2011]

1 2A, 3π4

2 A, 4π3

3 3A, 5π6

4 A, π3

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1 $\sqrt{2} A, \frac{3 π}{4}$

2 $A, \frac{4 π}{3}$

3 $\sqrt{3} A, \frac{5 π}{6}$

4 $A, \frac{π}{3}$