Question 1: Oscillations

Question

Five identical springs are used in the three configurations as shown in figure. The time periods of vertical oscillations in configurations (a), (b) and (c) are in the ratio.

Accepted Answer

To determine the time periods of vertical oscillations in different configurations of springs, we examine each setup.
Configuration (a)
:
The time period, $ T_a $, is given by the formula:
$    T_{a} = 2\pi \sqrt{\frac{m}{k}}    $
Configuration (b)
:
In this setup, two identical springs are arranged in series, effectively halving the spring constant. Thus, the time period, $ T_b $, is:
$    T_{b} = 2\pi \sqrt{\frac{m}{\frac{k}{2}}} = 2\pi \sqrt{\frac{2m}{k}}    $
Configuration (c)
:
In this setup, two identical springs are arranged in parallel, effectively doubling the spring constant. Hence, the time period, $ T_c $, is:
$    T_{c} = 2\pi \sqrt{\frac{m}{2k}}    $
Now, calculating the ratios of these time periods, we get:
$ T_{a} : T_{b} : T_{c} = 1 : \sqrt{2} : \frac{1}{\sqrt{2}} $
This ratio expresses the relative oscillation times for each configuration of the springs.

Answer

$ 1: \sqrt{2}: \frac{1}{\sqrt{2}} $

Answer

$ 2: \sqrt{2}: \frac{1}{\sqrt{2}} $

Answer

$ \frac{1}{\sqrt{2}}: 2: 1 $

Answer

$ 2: \frac{1}{\sqrt{2}}: 1 $

Oscillations

Five identical springs are used in the three configurations as shown in figure. The time periods of vertical oscillations in configurations (a), (b) and (c) are in the ratio.

Choose the correct answer: