91Ó°ÊÓ

A pulse travelling on a string is represented by the function $$ y=\frac{a^{3}}{(x-v t)^{2}+a^{2}} $$ where \(a=5 \mathrm{~mm}\) and \(v=20 \mathrm{~cm} \mathrm{~s}^{-1} .\) Sketch the shape of the string at \(t=0,1 \mathrm{~s}\) and \(2 \mathrm{~s}\). Take \(x=0\) in the middle of the string.

A turn of radius \(20 \mathrm{~m}\) is banked for the vehicles going at a speed of \(36 \mathrm{~km} / \mathrm{h}\). If the coefficient of static friction between the road and the tyre is \(0.4\), what are the possible speeds of a vehicle so that it neither slips down nor skids up ?

A particle is projected with a speed \(u\) at an angle \(\theta\) with the horizontal. Consider a small part of its path near the highest position and take it approximately to be a circular arc. What is the radius of this circle? This radius is called the radius of curvature of the curve at the point.

Three resonant frequencies of a string are 90,150 and \(210 \mathrm{~Hz} .\) (a) Find the highest possible fundamental frequency of vibration of this string. (b) Which harmonics of the fundamental are the given frequencies ? (c) Which overtones are these frequencies ? (d) If the length of the string is \(80 \mathrm{~cm}\), what would be the speed of a transverse wave on this string ?

The equation of a standing wave, produced on a string fixed at both ends, is $$ y=(0 \cdot 4 \mathrm{~cm}) \sin \left[\left(0-314 \mathrm{~cm}^{-1}\right) x\right] \cos \left[\left(600 \pi \mathrm{s}^{-1}\right) t\right] $$ What could be the smallest length of the string?