Oscillations

Question : 9 of 25

Marks: +1, -0

A spring having a spring constant 1200 N m

^{-1}

is mounted on a horizontal table as shown in figure. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.

Determine (i) the frequency of oscillations of the mass, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.

Solution:

Here, m = 3.0 kg ; k = 1200 N m

^{-1}

, A = 2 cm = 0.02 m

(i) The frequency of oscillations of the attached mass is

\upsilon = \frac{1}{2\pi} \sqrt{\frac{k}{m}}

= \frac{1}{2 \times 3.14} \sqrt{\frac{1200}{3}} = 3.2\,\text{s}^{-1}

(ii) The maximum acceleration of the mass is

|a_{\text{max}}| = \omega^{2} A = \frac{k A}{m} \left(\because \omega = \sqrt{\frac{k}{m}}\right)

= \frac{1200 \times 0.02}{3} = 8\,\mathrm{m}\,\mathrm{s}^{-2}

(iii) The maximum speed of the mass is

v_{\text{max}} = A \omega = A \sqrt{\frac{k}{m}}

= 0.02 \times \sqrt{\frac{1200}{3}} = 0.4\,\mathrm{m}\,\mathrm{s}^{-1}

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