Question

A spring produces extension $x$ by applying a force $F$. A body of mass $m$ suspended from the spring oscillates vertically with a period $T$. The mass of the suspended body is (neglect mass of spring)

• A. $\dfrac{2T^2F}{\pi^2 x}$
• B. $\dfrac{T^2F}{4\pi^2 x}$
• C. $\dfrac{T^2F}{\pi^2 x}$
• D. $\dfrac{T^2F}{2\pi^2 x}$

Hint:

Always use Hooke’s law first to find spring constant before applying oscillation formulas.

Answer 1

The Correct Option is B

Step 1: Find spring constant using Hooke’s law.
According to Hooke’s law: \[ F = kx \Rightarrow k = \dfrac{F}{x} \] Step 2: Write time period formula for spring-mass system.
The time period of vertical oscillation is: \[ T = 2\pi \sqrt{\dfrac{m}{k}} \] Step 3: Substitute value of spring constant.
\[ T = 2\pi \sqrt{\dfrac{mx}{F}} \] Step 4: Solve for mass $m$.
\[ T^2 = 4\pi^2 \dfrac{mx}{F} \] \[ m = \dfrac{T^2F}{4\pi^2 x} \]