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A spring having with a spring constant $1200 \mathrm{Nm}^{-1}$ is mounted on a horizontal table as shown in Fig. A mass of $3 \mathrm{~kg}$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $2.0 \mathrm{~cm}$ and released.
Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.
Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.
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Given, $k=1200 \mathrm{~N} / \mathrm{m} ; m=3 \mathrm{~kg} ; a=2 \mathrm{~cm}=0.02 \mathrm{~m}$
(i) Frequency $=v=\frac{1}{T}=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}$
$$
=\frac{1}{2 \times 3.14} \sqrt{\frac{1200}{3}}=3.2 \mathrm{~s}^{-1}
$$
(ii) Acceleration $=\omega^2 y=\frac{k}{m} y$;
Max. acceleration $=\frac{k a}{m}=\frac{1200 \times 0.02}{3}=8 \mathrm{~m} / \mathrm{s}^2$
(iii) Max. speed $=a \omega$
$$
=a \sqrt{\frac{k}{m}}=0.02 \times \sqrt{\frac{1200}{3}}=0.4 \mathrm{~m} / \mathrm{s}
$$
(i) Frequency $=v=\frac{1}{T}=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}$
$$
=\frac{1}{2 \times 3.14} \sqrt{\frac{1200}{3}}=3.2 \mathrm{~s}^{-1}
$$
(ii) Acceleration $=\omega^2 y=\frac{k}{m} y$;
Max. acceleration $=\frac{k a}{m}=\frac{1200 \times 0.02}{3}=8 \mathrm{~m} / \mathrm{s}^2$
(iii) Max. speed $=a \omega$
$$
=a \sqrt{\frac{k}{m}}=0.02 \times \sqrt{\frac{1200}{3}}=0.4 \mathrm{~m} / \mathrm{s}
$$
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