Ncert Exercises & Solutions

When a mass \(m\) is connected individually to two springs \(S_1\) and \(S_2,\) the oscillation frequencies are \(\nu_1\) and \(\nu_2.\) If the same mass is attached to the two springs as shown in the figure, the oscillation frequency would be:

1. \(v_2+v_2\)

2. \(\sqrt{v_1^2+v_2^2}\)

3. \(\frac{1}{v_1}+\frac{1}{v_1}^{-1}\)

4. \(\sqrt{v_1^2-v_2^2}\)

2. Hint: Two springs are in parallel.

Step 1: Find the equivalent frequency of the system.

Consider the diagram,

k

=Equivaent spring constant

= k_{1} + k_{2}

The time period of oscillation of the spring block-system,

T = 2 π \sqrt{\frac{m}{k_{eq}}} = 2 π \sqrt{\frac{m}{k_{1} + k_{2}}}

\Rightarrow v = \frac{1}{T} = \frac{1}{2 π} \times \sqrt{\frac{k_{1} + k_{2}}{m}}

...(i)

Step 2: Find the individual frequencies.

When the mass is connected to the springs individually:

v_{1} = \frac{1}{2 π} \sqrt{\frac{k_{1}}{m}} . . . (i i)

v_{2} = \frac{1}{2 π} \sqrt{\frac{k_{2}}{m}} . . . (i i i)

Using equations (i), (ii) and (iii)....

v = \frac{1}{2 π} \times \sqrt{4 π^{2} {ν_{1}}^{2} + 4 π^{2} {ν_{2}}^{2}} = \sqrt{{ν_{1}}^{2} + {ν_{2}}^{2}}

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