Investigation 10C: Springs and the conservation of energy
Can energy conservation be used to predict the behavior of a system?
A system consisting of a cart, attached to a spring, rolling freely on an inclined track includes three forms of energy. By carefully modeling the flow of energy among potential, kinetic, and elastic forms, we can accurately predict how far down the track the cart will roll before it is stopped by the spring.
Building a mathematical model and testing it
Set up your equipment as in the picture with two 250-g cart masses on the cart.
Given the setup, derive an expression which predicts the maximum distance the cart rolls down the track after being released from the top of the track (top is where the spring is unstretched).
Write your expression in terms of spring extension x, mass m, spring constant k, and gravitational acceleration g.
NOTE: Because the track is inclined 30° a helpful substitution can be made for height: h = x/2.
Measure the mass of the cart plus masses, and measure the spring constant of the spring. Use those values and your expression to predict the maximum distance the cart will roll down the track.
Open the file 10C_EnergyTransformations, and then connect your Smart Cart using Bluetooth.
Hold the cart at the top of the track (where the spring is unstretched), start data collection, and then release the cart. Catch the cart at the top of its rebound.
How does your experimental value for maximum distance dexp compare to your prediction? What may have caused differences between them?
Use your experimental value for dexp, your motion data, and energy equations to complete the table.
How did the total energy of the cart and spring system change as the cart rolled down the track?
Create a bar chart to display the kinetic, gravitational potential, elastic potential, and total energies at each of the four positions shown in the table. What pattern do you see?
Calculate the efficiency of the system for two scenarios:
From the point of release to the furthest point reached on the track
From the point of release to the rebound position when the cart was caught.
Measuring spring constant
Set up the apparatus like the picture. Make sure the apparatus is level.
Open the 10C_SpringConstant experiment file. Power on the smart cart and connect it using Bluetooth.
In your software zero the smart cart force sensor while nothing is touching the hook.
Attach the spring to the hook and position the cart so that it is about to place the spring under tension.
Record data as you slowly pull the cart about 70 cm along the track. Data collection will stop after 4 s.
Find the slope of the straight line portion of the graph. This is k, the spring constant for the spring.