First slide

Simple hormonic motion

Question

A block is suspended by a spring and allowed to oscillate vertically with small amplitude. Time period of oscillation is T. Now the spring is cut into two parts whose lengths are in the ratio 2 :3. The two parts of the spring are attached to the same block as shown in figure. The block is now slightly pushed to the right and released. The new time period of oscillation is

Moderate

A

$\frac{6 T}{5}$
B

$\frac{\sqrt{6} T}{5}$
C

$5 T$
D

$3 T$

Solution

In the first case, $T = 2 π \sqrt{\frac{m}{K}}$
In the second case, $K_{1} = (\frac{2 + 3}{2}) K = \frac{5 K}{2} a n d K_{2} = (\frac{2 + 3}{3}) K = \frac{5 K}{3}$
$∴ K_{e f f} = K_{1} + K_{2} = \frac{25 K}{6}$
$∴ T' = 2 π \sqrt{\frac{m}{K_{e}}} = 2 π \sqrt{\frac{m}{25 K / 6}} = \frac{\sqrt{6}}{5} T$

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