1. Relation for spring constant

For a mass-spring system, the time period is

$T = 2\pi \sqrt{\frac{m}{k}}$

Squaring both sides:

$T^2 = 4\pi^2 \frac{m}{k}$

So,

$k = 4\pi^2 \frac{m}{T^2}$

Hence,

$k \propto \frac{m}{T^2}$

2. Use percentage error formula

For a quantity

$k \propto m\, T^{-2}$

the fractional error is

$\frac{\Delta k}{k} = \frac{\Delta m}{m} + 2\frac{\Delta T}{T}$

when taking maximum percentage error in magnitude.

Given:

• Mass has 1% negative error $\Rightarrow$ magnitude of error in $m$ is $1\%$
• Time has 2% positive error $\Rightarrow$ magnitude of error in $T$ is $2\%$

Therefore,

$\%\text{ error in }k = 1\% + 2(2\%) = 1\% + 4\% = 5\%$

3. Check options

• A: $5\%$ ✅
• B: $3\%$
• C: $1\%$
• D: $4\%$

So the correct option is A.

4. Comparison with stored answer

Stored correct answer: A

My derived answer: A

They match.