Question 6: Oscillations

Question

Time period of oscillation of a mass suspended from a spring is T . If the spring is cut into four equal parts and the same mass is suspended from one of the parts, the fractional change in time period is

Accepted Answer

Let the original spring constant be $k$ and the mass be $m$. Then
$ T \;=\; 2\pi\sqrt{\frac{m}{k}}\,.   $
If you cut the spring into 4 equal parts, each piece has spring constant
$ k' \;=\; 4k   $
because for $n$ equal pieces $k'\!=\!n\,k$. The new period is
$ T' \;=\; 2\pi\sqrt{\frac{m}{k'}} \;=\; 2\pi\sqrt{\frac{m}{4k}}       \;=\; \frac{1}{2}\;2\pi\sqrt{\frac{m}{k}}       \;=\; \frac{T}{2}\,.   $
The fractional change in period is
$ \frac{T' - T}{T} \;=\; \frac{	frac{T}{2} - T}{T} \;=\; -\frac{1}{2}\,, $
i.e. a decrease by $	frac12$.
Answer: Option C, $	frac12$.

Answer

$\frac{1}{8}$

Answer

$\frac{1}{4}$

Answer

$\frac{1}{2}$

Answer

$\frac{1}{6}$

Oscillations

Time period of oscillation of a mass suspended from a spring is T . If the spring is cut into four equal parts and the same mass is suspended from one of the parts, the fractional change in time period is

Choose the correct answer: