Physics - SHM Question with Solution | TestHub

$\text{T}=2\text{π}\sqrt{\frac{\text{m}+6}{\text{600}}}\mathrm{ \; \; \; \; \; \; \; \; }(\text{T}=2\text{π}\sqrt{\frac{\text{m}}{\text{k}}})$

$\text{or}\text{0.75}=2\text{π}\sqrt{\frac{\text{m}+6}{\text{600}}}$

$\therefore \text{m}=\frac{{\left(\text{0.75}\right)}^2\times \text{600}}{{\left(2\text{π}\right)}^2}-6$

= 2.55 kg

Maximum acceleration of SHM is,

a_max= ω²A   (A = amplitude)

i.e., maximum force on mass 'm' is m ω²A which is being provided by the force of friction between the mass and the cart. Therefore,

μ_smg  ≥  mω²A

$\mathrm{or\; \; \; \; \; }{\text{μ}}_{\text{s}}\ge \frac{{\text{ω}}^2\text{A}}{\text{g}}$

$\mathrm{or\; \; \; \; \; }{\text{μ}}_{\text{s}}\ge (\frac{2\text{π}}{\text{T}}{)}^2·\frac{\text{A}}{\text{g}}$

$\mathrm{or\; \; \; \; \; }{\text{μ}}_{\text{s}}\ge \left(\frac{2\text{π}}{\text{0.75}}{)}^2\right(\frac{\text{0.05}}{\text{9.8}})\left(\text{A = 50 mm}\right)$

or               μ_s≥  0.358

Thus, the minimum value of  μ_sshould be 0.358.