Sheet 01 Harmonic Functions and Simple Harmonic Motion.
Which of the following functions of time represents (a) Periodic and (b) Non-periodic motion. Give the period for each case of periodic motion. (\(\omega\) is any positive constant). (i) \(\left(Sin\omega t+Cos\omega t\right)\). (ii) \(\left(Sin\omega t+ Cos2\omega t+ Sin4\omega t\right)\). (iii) \(e^{-\omega t}\). (iv) \(log\left(\omega t\right)\).
A particle executes S.H.M. given by \(y=0.24Sin\left(400t+0.5\right)\) in S.I units. Find: (i) Amplitude. (ii) Frequency. (iii) Time period of the vibration. (iv) initial phase.
Ans:(i) \(a=0.24m\), (ii) \(f=\frac{200}{\pi}hertz\) and (iii) \(T=0.0157s\) (iv) \(\phi_{o}=0.5rad\).
A particle executing S.H.M. with time period of \(4s\) and amplitude \(6cm\). Calculate: (i) Displacement (ii) Velocity (iii) Acceleration of the particle after \(1s\) starting from the mean position.
Ans:(i) \(Y=5cm\), (ii) \(V=0(Zero)\) and (iii) \(a=\frac{5}{4}\pi^{2}cm.s^{-2}\).
A body oscillates with S.H.M. according to the equation \(x(t)=5Cos\left(2\pi t+\frac{\pi}{4}\right)\). Where; \(t\) is in seconds and \(x\) is in meters. Calculate: (a) Displacement at \(t=0sec\), (b) Time period. (c) Initial Velocity.
Ans:(a) \(x_{t=0}=\frac{5}{\sqrt{2}}m\). (b) \(T=1Sec\) and (c) \(V_{t=0}=-\frac{10\pi}{\sqrt{2}}\frac{m}{s}\).
A body oscillates with S.H.M. according to the following equation \(x=(5.0m)Cos\left[\left(2\pi\frac{rad}{s}\right).t+\frac{\pi}{4}\right]\) at \(t=1.5Sec\). Calculate: (a) Displacement (b) Speed and (c) Acceleration of the body.
Ans:(a) \(x=-3.535m\), (b) \(v=22.22\frac{m}{s}\) and (iii) \(a=139.56\frac{m}{s}\).
A particle is moving with straight line has velocity \(v\) given by \(v^{2}=\alpha-\beta y^{2}\), where; \(\alpha\) and \(\beta\) are constants and \(y\) is the distance from the fixed point in the line. Show that the motion of particle is S.H.M. Also; find its time period and amplitude.
Ans: \(T=\frac{2\pi}{\sqrt{\beta}}\) and \(r=\sqrt{\frac{\alpha}{\beta}}\).
Two simple harmonic motions are represented by the following equations (i) \(y_{1}=10Sin\left(3\pi t+\frac{\pi}{4}\right)\), and (ii) \(y_{2}=5\left(Sin3\pi t+\sqrt{3}Cos3\pi t\right)\). Find: the ratio of their amplitudes.
Ans: \(\frac{a_{1}}{a_{2}}=1\).
A particle executes S.H.M. on a straight line path. The amplitude of oscillation is \(2cm\). When the displacement of the particle from the mean position is \(1cm\), Find the (a) Time period. (b) Maximum velocity and (c) Maximum acceleration of S.H.M.
Ans:(a) \(T=3.63s\), (b) \(v_{max}=3.464\frac{m}{s}\) and (c) \(a_{max}=6\frac{m}{s^{2}}\).
For a particle is S.H.M. the displacement \(x\) of the particle as a function of time \(t\) is given by; \(x=rSin(2\pi t)\). Here; \(x\) is in cm and \(t\) is in Seconds. Let the time taken by the particle travel from \(x=0\) to \(x=\frac{r}{2}\) be \(t_{1}\) and the time taken to travel from \(x=\frac{r}{2}\) to \(x=r\) be \(t_{2}\). Find: \(\frac{t_{1}}{t_{2}}\).
Ans: \(\frac{t_{1}}{t_{2}}=\frac{1}{2}\).
The displacement of a particle executing periodic motion is given by; \(y=4Cos^{2}\left(\frac{1}{2}t\right)Sin(100t)\). Find: the independent constituent S.H.M.
Ans: \(y=2Sin(100t)+Sin(101t)+Sin(99t)\).
A block of mass one kg is fastened to a spring with a spring constant \(50\frac{N}{m}\). The block is pulled to a distance \(x=10cm\) from its equilibrium position at \(x=0\) on a frictionless surface from rest at \(t=0\). Write the expression for its \(x(t)\) and \(v(t)\).
Ans: \(x(t)=0.10Sin(7.07t)m\) and \(v(t)=0.707Cos(7.07t)\frac{m}{s}\).
A man stands on a weighing machine placed on a horizontal platform. The machine reads \(50kg\), by means of a suitable mechanism the platform is made to execute harmonic vibrations up and down with a frequency of 2 vibrations per second. What will be the effect on the reading of the weighing machine? The amplitude of the vibration of the weighing machine is \(5cm\). Take: \(g=10\frac{m}{s^{2}}\).
Ans: \(W’=10.5kgf-89.5kgf\).
A particle executes S.H.M. along a straight line. Its velocity is \(4\frac{m}{s}\) when it is \(3m\) from the mean position and \(3\frac{m}{s}\) when it is at a distance of \(4m\) from it. Find: the time it takes to travel \(2.5m\) from the positive extrimity of its oscillation.
