Question

4.Three SHM'sof equal amplitude A and equal time period in the same direction combine.The difference in phase between each pair is 60degrees a head of the other.The amplitude of the resultant oscillation is ----------

Answer 1

$$\begin{gathered} \mathrm{Resultant}\mathrm{of}\mathrm{first}\mathrm{two}\mathrm{SHMs}: \\ {\mathrm{R}}_{12}=\sqrt{{{\mathrm{R}}_1}^2+{{\mathrm{R}}_2}^2+2{\mathrm{R}}_1{\mathrm{R}}_2\mathrm{cos\theta }} \\ =\sqrt{{\mathrm{A}}^2+{\mathrm{A}}^2+2\left(\mathrm{A}\right)\left(\mathrm{A}\right)\cos {60}^{\mathrm{o}}} \\ =\sqrt{3}\mathrm{A} \\ \mathrm{Angle},\mathrm{\phi }=\frac{\mathrm{\theta }}{2} \\ =\frac{{60}^{\mathrm{o}}}{2} \\ ={30}^{\mathrm{o}} \\ \mathbf{Resultant}\mathbf{}\mathbf{of}\mathbf{}\mathbf{all}\mathbf{}\mathbf{three}\mathbf{:} \\ \mathrm{It}\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{resultant}\mathrm{of}\mathrm{first}\mathrm{two}\mathrm{and}\mathrm{then}\mathrm{the}\mathrm{third}\mathrm{as}\mathrm{follows}: \\ \mathrm{Angle}\mathrm{between}{\mathrm{R}}_{12}\mathrm{and}{\mathrm{R}}_3,\mathrm{\delta }=60+30={90}^{\mathrm{o}} \\ \mathrm{R}=\sqrt{{{\mathrm{R}}_{12}}^2+{{\mathrm{R}}_3}^2+2{\mathrm{R}}_{12}{\mathrm{R}}_3\cos \mathrm{\delta }} \\ =\sqrt{{\left(\sqrt{3}\mathrm{A}\right)}^2+{\left(\mathrm{A}\right)}^2+2\left(\sqrt{3}\mathrm{A}\right)\left(\mathrm{A}\right)\cos {90}^{\mathrm{o}}} \\ \mathbf{=}\mathbf{2}\mathbf{A} \end{gathered}$$