Quick answer: Work done by a variable force is the integral W = ∫F dx, the area under the force–position graph. This AP Physics C lab on work energy theorem gives the aim, the equipment, the safety points, a step-by-step method, sample results and 7 analysis questions; the answers are on a separate page and the lab sheet is a free printable PDF.
A spring stores energy as ½kx². In this lab you measure k by adding weights slowly, find the stored energy from the area under the force–extension graph, then drop a mass from the unstretched position and test whether the greatest extension equals 2mg/k, as energy conservation predicts.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 3
Aim
Measure the spring constant from a force–extension graph, find the elastic energy as the area under the graph, and test the prediction that a mass released from the unstretched position falls a distance 2mg/k.
Equipment
- Light helical spring (k about 25 N/m)
- Slotted masses, 50 g steps up to 250 g, and a hanger
- Retort stand, boss and clamp
- Metre rule fixed beside the spring
- Phone for slow-motion video
Safety
Place a soft pad on the floor under the masses and keep feet away from the drop zone. Wear goggles in case the spring slips off the hook when a mass is released.
Method
- Hang the spring and record the position of the bottom of the spring with no mass.
- Add 50 g at a time and record the extension x after each addition, once the mass is still.
- For the drop test, hold a 0.100 kg mass at the unstretched position, release it, and film the fall beside the rule. Find the greatest extension from the video, three times.
- Use F = mg with g = 9.81 m/s² for each load.
Results
| Mass m (g) | Force F (N) | Extension x (cm) |
|---|---|---|
| 50 | 0.49 | 2.0 |
| 100 | 0.98 | 3.9 |
| 150 | 1.47 | 5.9 |
| 200 | 1.96 | 7.9 |
| 250 | 2.45 | 9.8 |
Drop test, 0.100 kg released from the unstretched position:
| Trial | Greatest extension (cm) |
|---|---|
| 1 | 7.7 |
| 2 | 7.5 |
| 3 | 7.6 |
Analysis questions
Question 1 Spring constant
Plot F against x in metres. Draw the best-fit line through the origin and find k from the gradient.Question 2 Area under the graph
Calculate the area under the line from x = 0 to x = 0.098 m and compare it with ½kx² for the same extension.Question 3 Predicted drop
Use mgx = ½kx² to predict the greatest extension for the 0.100 kg mass.Question 4 Measured against predicted
Find the mean measured extension and the percentage difference from your prediction.Question 5 Energy accounting
At the measured greatest extension, calculate the gravitational energy lost and the elastic energy stored. Find the energy unaccounted for.Question 6 Uncertainty
The extension is read to ±1 mm. Calculate the percentage uncertainty in x for the 50 g reading and say how it affects k.Question 7 Evaluation
Give two reasons why the measured extension is smaller than the prediction and one improvement.
