Question

What is the mass of the ball that was hung to the spring if the period is 4 sec and the spring constant is 80 N/m.

Answer 1

Approximately 32 kg

Step-by-step explanation:

What we know are period (T) = 4 seconds and spring constant (k) = 80 N/m (Newton/meter)

Therefore, from the formula of:

`\displaystyle{T = 2\pi \sqrt{ (m)/(k)} }`

Substitute in known values:

`\displaystyle{4 \: \text{seconds} = 2\pi \sqrt{ \frac{m}{80 \: \text{N/m}} }}`

Divide both sides by 2π:

`\displaystyle{ \frac{4 \: \text{seconds} }{2\pi}=\sqrt{ \frac{m}{80 \: \text{N/m}} }} \\ \\ \displaystyle{ \frac{2 \: \text{seconds} }{\pi}=\sqrt{ \frac{m}{80 \: \text{N/m}} }}`

Square both sides:

`\displaystyle{ \left( \frac{2 \: \text{seconds} }{\pi} \right)^(2) = \left(\sqrt{ \frac{m}{80 \: \text{N/m}} } \right)^(2) } \\ \\ \displaystyle{ \frac{4 \: \text{s}^2 }{ {\pi}^(2) } = \frac{m}{80 \: \text{N/m}}}`

Then move 80 N/m to multiply left side:

`\displaystyle{ \frac{4 \: \text{s}^2 }{ {\pi}^(2) } \cdot 80 \: \text{N/m}= m} \\ \\ \displaystyle{ \frac{320 }{ {\pi}^(2) } \: \text{kg}= m}`

Since you didn't specify which digit of π to use so I will go with what most people use, π ≈ 3.14

`\displaystyle{ \frac{320}{ {(3.14)}^(2) } \: \text{kg} = m}`

3.14*3.14 = 9.8596 but rounded to 9.86 through significant figure rule:

`\displaystyle{ (320)/(9.86) \: \text{kg} = m}`

320 can be converted to scientific notation which is 3.2*10². Therefore, this has 2 figures. Hence, the result must be in 2 digits after division:

`\displaystyle{32 \approx m}`

Therefore, the mass is around 32 kg.