Question

The force \(F\) required to compress a spring a distance \(x\) is given by \(F-F_{0}=k x\) where \(k\) is the spring constant and \(F_{0}\) is the preload. Determine the work required to compress a spring whose spring constant is \(k=200\) lbf/in a distance of one inch starting from its free length where \(F_{0}=0\) lbf. Express your answer in both lbf:ft and Btu.

Short answer

Answer: The work required to compress the spring is 8.33 lbf*ft or 0.0107 Btu.

Step-by-step solution

Write down the given information

The given information is: Spring constant: \(k=200\) lbf/in Displacement: \(x=1\) in Initial force: \(F_0=0\) lbf

Write down the formula for force

The formula for force is given by: \(F - F_0 = kx \)

Substitute the given values into the force formula

Substituting the given values, we have: \(F - 0 = 200(1)\) Therefore, \(F = 200\) lbf

Calculate the work done

The formula for work is given by the integral of force with respect to distance: \(W = \int_{0}^{x} F dx\) In this case, since the force is linearly proportional to the distance, the work can be calculated as the area under the force-displacement graph (a triangle): \(W = \frac{1}{2}(F)(x) = \frac{1}{2}(200 \text{ lbf})(1 \text{ in}) = 100 \text{ lbf}* \text{in}\)

Convert work to necessary units

We need to express our answer in lbf*ft and Btu. To convert lbf*in to lbf*ft: \(100 \text{ lbf}* \text{in} \times \frac{1 \text{ ft}}{12 \text{ in}} = 8.33 \text{ lbf}* \text{ft}\) To convert lbf*ft to Btu, we can use the conversion factor: \(1 \text{ Btu} = 778.169 \text{ lbf}* \text{ft}\) So, \(8.33 \text{ lbf}* \text{ft} \times \frac{1 \text{ Btu}}{778.169 \text{ lbf}* \text{ft}} = 0.0107 \text{ Btu}\)

Final Answer

The work required to compress the spring is \(8.33\) lbf*ft or \(0.0107\) Btu.