Each end of the cart shown above is attached via a spring to the corresponding end of the track. When you displace the cart toward either end of the track and release it, it oscillates about the middle of the track. (The two springs have equal spring constants and free lengths.) You can discuss the work you do in displacing the cart, its conversion into potential energy in the springs, the work the springs do on the cart in converting that potential energy into kinetic energy in the cart as they accelerate it toward the middle of the track, the work the cart does on the springs as it goes from the middle of the track to the turning point and converts its kinetic energy back into potential energy in the springs, and how this process repeats as the cart oscillates back and forth. As shown in the photograph, the cart is carrying two iron bars, each of which has a mass of 500 g. The cart itself has a mass of 500 g, so you can choose to operate the apparatus with the cart having a mass of 500 g, 1 kg or 1.5 kg.

In the apparatus in the photograph above, a cart rides in a pair of grooves on a track. Each end of the cart is connected to the corresponding end of the track via a spring. The two springs are of equal free length and have equal spring constants. When the system is in equilibrium, as shown in the photograph, both springs are extended and apply equal and opposite forces to the cart. The net force on the cart is zero, and it sits at the middle of the track.

When you extend or compress a spring, the force you exert to do so is F = kx, where x is the distance by which you have extended or compressed the spring, relative to its original length, and k is the force constant of the spring. k is a measure of the stiffness of the spring, and its units are N/m. The force the spring exerts in turn is F = -kx, which opposes the force you apply. This is Hooke’s Law.

When you displace the cart from the middle by some distance x, you extend one spring further while compressing the other (or allowing it to relax by the distance x). The force exerted by the springs to oppose the force you apply is the sum of the changes in the force exerted by each spring. The effective force constant of the two springs together is thus the sum of the two individual force constants. Since both springs have the same force constant (about 3.3 N/m), the force constant for the system is twice this (about 6.6 N/m).

The work you do when you displace the cart is W = √F · dx (=√kx dx), or W = (1/2)kx². This is also the potential energy that is stored in the springs when you do this. When you release the cart, the springs now do work on the cart, equal to the potential energy that was stored in them. As the cart accelerates toward the equilibrium position, the potential energy in the springs is converted into kinetic energy of the cart, until the cart reaches the equilibrium position. At this point, all of the potential energy has been converted into kinetic energy; the potential energy equals zero, and the kinetic energy is K = (1/2)kx². As the cart continues past the equilibrium position, it does work on the springs, slowing as its kinetic energy is converted back into potential energy in the springs. When its kinetic energy is zero, the cart stops, and all of the energy is now potential energy in the springs. The springs now do work on the cart, accelerating it back toward the equilibrium position, and the process continues. The cart oscillates back and forth along the track.

We can write the equation above, F = -kx, as m(d²x/dt²) + kx = 0, which is a second-order differential equation that describes the motion of a simple harmonic oscillator. Its solution is described on the pages for demonstration 40.06 -- Cart between springs, and for demonstration 40.12 -- Mass-springs with different spring constants and masses.