A uniform rope of length \(L\) and mass \(m_1\) hangs vertically from a rigid support. A block of mass \(m_2\) is attached to the free end of the ropes. A transverse pulse of wavelength \(\lambda_1\) is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is \(\lambda_2\). The ratio \(\dfrac{\lambda_2}{\lambda_1}\) is:
1. \(\sqrt{\dfrac{m_1+m_2}{m_1}}\)
2. \(\sqrt{\dfrac{m_2}{m_1}}\)
3. \(\sqrt{\dfrac{m_1+m_2}{m_2}}\)
4. \(\sqrt{\dfrac{m_1}{m_2}}\)

If n₁, n₂ and n₃ are, are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency n of the string is given by

1. 1/n=1/n₁+1/n₂+1/n₃
 

2. 1/√n=1/√n₁+1/√n₂+1/√n₃

3. √n=√n₁+√n₂+√n₃

4. n=n₁+n₂+n₃

When a string is divided into three segments of lengths $l_1,<mspace\; width="1em"></mspace>l_2<mspace\; width="1em"></mspace>and<mspace\; width="1em"></mspace>l_3,$ the fundamental frequencies of these three segments are $v_1,<mspace\; width="1em"></mspace>v_2<mspace\; width="1em"></mspace>and<mspace\; width="1em"></mspace>v_3$ respectively. The original fundamental frequency (v) of the string is 

1. $\sqrt{v}=\sqrt{v_1}+\sqrt{v_2}+\sqrt{v_3}$

2. $v=v_1+v_2+v_3$

3. $\frac{1}{v}=\frac{1}{v_1}+\frac{{\displaystyle 1}}{{\displaystyle v_2}}+\frac{{\displaystyle 1}}{{\displaystyle v_3}}$

4. $\frac{1}{\sqrt{v}}=\frac{1}{\sqrt{v_1}}+\frac{{\displaystyle 1}}{{\displaystyle \sqrt{v_2}}}+\frac{{\displaystyle 1}}{{\displaystyle \sqrt{v_3}}}$

A wave in a string has an amplitude of 2 cm. The wave travels in the +ve direction of x-axis with a speed of $128<mspace\; width="1em"></mspace>{\mathrm{ms}}^{-1}$ and it is noted that 5 complete waves fit in 4 m length of the string. The equation describing the wave is :

1. $y=\left(0.02\right)\mathrm{m}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>\left(7.85\mathrm{x}<mspace\; width="1em"></mspace>+1005\mathit{\hspace{0.17em}}t<mspace\; width="1em"></mspace>\right)$

2. $y=\left(0.02\right)\mathrm{m}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>\left(15.7\mathrm{x}<mspace\; width="1em"></mspace>-2010\mathit{\hspace{0.17em}}t<mspace\; width="1em"></mspace>\right)$

3. $y=\left(0.02\right)\mathrm{m}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>\left(15.7\mathrm{x}<mspace\; width="1em"></mspace>+2010\mathit{\hspace{0.17em}}t<mspace\; width="1em"></mspace>\right)$

4. $y=\left(0.02\right)\mathrm{m}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>\left(7.85\mathrm{x}<mspace\; width="1em"></mspace>-1005\mathit{\hspace{0.17em}}t<mspace\; width="1em"></mspace>\right)$

The equation of a wave traveling in a string can be written as $y=3\cos \pi (100t-x)$. Its wavelength is :

(1) 100 cm

(2) 2 cm

(3) 5 cm

(4) None of the above