A $S.H.M.$ is represented by $x = 5\sqrt 2 (\sin 2\pi t + \cos 2\pi t).$ The amplitude of the $S.H.M.$ is .... $cm$
Medium
Download our app for free and get started
(a) $x = 5\sqrt 2 \left( {\sin 2\pi \,t + \cos 2\pi \,t} \right)$
$ = 5\sqrt 2 \sin 2\pi \,t + 5\sqrt 2 \cos 2\pi \,t$
$x = 5\sqrt 2 \sin 2\pi t + 5\sqrt 2 \sin \,\left( {2\pi \,t + \frac{\pi }{2}} \right)$
Phase difference between constituent waves $\phi = \frac{\pi }{2}$
$\therefore $ Resultant amplitude $A = \sqrt {{{(5\sqrt 2 )}^2} + {{(5\sqrt 2 )}^2}} \,=10 \,cm.$
$ = 5\sqrt 2 \sin 2\pi \,t + 5\sqrt 2 \cos 2\pi \,t$
$x = 5\sqrt 2 \sin 2\pi t + 5\sqrt 2 \sin \,\left( {2\pi \,t + \frac{\pi }{2}} \right)$
Phase difference between constituent waves $\phi = \frac{\pi }{2}$
$\therefore $ Resultant amplitude $A = \sqrt {{{(5\sqrt 2 )}^2} + {{(5\sqrt 2 )}^2}} \,=10 \,cm.$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1A particle executes simple harmonic motion between $x =- A$ and $x =+ A$. If time taken by particle to go from $x=0$ to $\frac{A}{2}$ is $2 s$; then time taken by particle in going from $x =\frac{ A }{2}$ to $A$ is $.........\,s$View Solution
- 2The energy of a particle executing simple harmonic motion is given by $E = Ax^2 + Bv^2$, where $x$ is the displacement from mean position $x = 0$ and $v$ is the velocity of the particle at $x$ then choose the incorrect statementView Solution
- 3A particle executing $SHM$ of amplitude $4\,cm$ and $T = 4s.$ The time taken by it to move from $+2\,cm$ to $+2\sqrt 3\,cm$ isView Solution
- 4A pendulum is hung from the roof of a sufficiently high building and is moving freely to and fro like a simple harmonic oscillator. The acceleration of the bob of the pendulum is $20\; m/s^2$ at a distance of $5\; m$ from the mean position. The time period of oscillation isView Solution
- 5In a seconds pendulum, mass of bob is $30\, gm$. If it is replaced by $90\, gm$ mass. Then its time period will .... $\sec$View Solution
- 6In an engine the piston undergoes vertical simple harmonic motion with amplitude $7\, cm$ . A washer rests on top of the piston and moves with it. The motor speed is slowly increased. The frequency of the piston at which the washer no longer stays in contact with the piston, is close to ...... $Hz$View Solution
- 7A particle of mass $m$ is dropped from a height $R$ equal to the radius of the earth above the tunnel dug through the earth as shown in the figure. Hence the correct statement isView Solution
- 8Two bodies of masses $1\, kg$ and $4\, kg$ are connected to a vertical spring, as shown in the figure. The smaller mass executes simple harmonic motion of angular frequency $25\, rad/s$, and amplitude $1.6\, cm$ while the bigger mass remains stationary on the ground. The maximum force exerted by the system on the floor is ..... $N$ ( take $g = 10\, ms^{-2}$)View Solution
- 9A particle of mass $m$ in a unidirectional potential field have potential energy $U(x)=\alpha+2 \beta x^2$, where $\alpha$ and $\beta$ are positive constants. Find its time period of oscillation.View Solution
- 10Force constant of a spring is $K$ . If half part is detached then force constant of the remaining spring will beView Solution