A mass ' $M$ ' is suspended from a light spring. An additional mass $\mathrm{M}_1$ added extends the spring…

A mass ' $M$ ' is suspended from a light spring. An additional mass $\mathrm{M}_1$ added extends the spring further by a distance ' $\mathrm{x}$ '. Now the combined mass will oscillate on the spring with period $\mathrm{T}=$
  1. $2 \pi\left[\left(\frac{M_1 g}{x\left(M+M_1\right)}\right)\right]^{\frac{1}{2}}$
  2. $2 \pi\left[\frac{\left(\mathrm{M}+\mathrm{M}_1\right) \mathrm{x}}{\mathrm{M}_1 \mathrm{~g}}\right]^{\frac{1}{2}}$
  3. $\left(\frac{\pi}{2}\right)\left[\left(\frac{M_1 g}{x\left(M+M_1\right)}\right)\right]^{\frac{1}{2}}$
  4. $\frac{\pi}{2}\left[\frac{\left(\mathrm{M}+\mathrm{M}_1\right) \mathrm{x}}{\mathrm{M}_1 \mathrm{~g}}\right]^{\frac{1}{2}}$

Solution

Time period $T=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$ Here, $k x=M_1 g$ $\therefore \quad \mathrm{K}=\frac{\mathrm{M}_1 \mathrm{~g}}{\mathrm{x}}$ $\therefore \quad$ Combined mass: $\mathrm{M}+\mathrm{M}_1$ $\therefore \quad \mathrm{T}=2 \pi \sqrt{\frac{\mathrm{x}\left(\mathrm{M}+\mathrm{M}_1\right)}{\mathrm{M}_1 \mathrm{~g}}}$ /

Asked in: MHT CET 2023 (13 May Shift 2)

Practice more Laws of Motion questions on Aicharya