Question

By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?

Answer 1

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

So that the ratio of the periods can be written in terms of their masses:

$$m_2=m_1\frac{T_1^2}{T_2^2}=0.5\frac{2. 01^2}{1.5^2}=0.8978\ kg\\\Delta m=0.898-0.889=0.009\ kg\\\frac{0.009}{0.389}=2\%$$

Similarly,

$$m_2=m_1\frac{T_1^2}{T_2^2}=0.5\frac{1.99^2}{1.5^2}=0.8978\ kg\\\Delta m=0.889-0.880=0.009\ kg\\\frac{0.009}{0.389}=2\%$$