- Parent Category: 05 Electricity and Magnetism
- Category: 08 Electromagnetism
- Created: Monday, 30 April 2018 15:03
- Last Updated: Monday, 30 April 2018 15:03
- Published: Monday, 30 April 2018 15:03
- Written by Fremont
- Hits: 8956
This is the simulation of the motion of a mass m situated at the end of a spring of length l and negligible mass. The motion is restricted to the horizontal dimension. (We choose a coordinate system in the plane with origin at the fixed end of the spring and with the X axis along the direction of the spring).
We assume that the reaction of the spring to a displacement dx from the equilibrium point can be modeled using Hooke's Law, F(dx) = -k dx , where k is a constant which depends on the physical characteristics of the spring. Thus, applying Newton's Second Law, we obtain the following second-order ordinary differential equation:
where x is the horizontal position of the free end of the spring.
In the simulation we solve numerically this equation and visualize the results.
- Measure the period of the motion for the given initial conditions.
- Drag with the mouse the ball to a new position and measure the period again. What do you observe?
- Set the mass of the ball to several different values (keeping k constant) and plot in your notebook the observed period versus the mass.
- Do the same for the elastic constant of the spring, k.
- Would you dare to provide an explicit formula for the dependence of the period with respect to the mass and k?
- Should the total energy of the model be preserved?
- Why do you think the total energy of the simulation slowly increases? (Hint: choose a better solver for the equations, such as Runge-Kutta, and check again.)
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Wolfgang Christian; Francisco Esquembre