The metal cylinder and the spring

In this project, we will look at a metal cylinder of mass m=5 kg with a hole in the middle and which is set on a metal rod. The cylinder slides horizontally along the rod. In addition, a spring is attached to the cylinder with a spring constant k=500N/m. When the spring is not stretched or compressed it has a length of l_0=0.5m. The spring is attached to a table placed below the metal cylinder and rod as shown in figure 1. The height between the table and the attachment point of the spring on the cylinder is h=0.3 m. We start by assuming that there is no friction or air resistance.

a) What is the horizontal position of the cylinder when the spring is at its equilibrium length l_0 = 0.5m? Define x = 0 m to be the point the spring is attached to the table, as shown in figure 1.

b) What is the length l of the spring for any given position x of the cylinder?

c) Draw a free body diagram of the cylinder. Remember to name all acting forces.

d) Show that the horizontal component of force from the spring on the cylinder is:

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Automatisk generert beskrivelse

e) Plot (using Python with numpy and matplotlib.pyplot) the horizontal spring force as a function of the position of the cylinder from x = -0.75 to x = +0.75. From the plot, explain how you expect the cylinder to move.

f) You drag the cylinder to a starting position x1 = 0.6 m and release it without giving any initial speed (you don’t push the cylinder; you only release it). Write a python program that solves the equations of motion numerically using the Euler-Cromer method and plot the position and velocity of the cylinder for the first 10 seconds of the movement. Describe and explain the movement.

e) Run the program again, but this time with starting position x1 = 0.65 m. Plot the position and velocity as a function of time for the first 10 seconds of the movement. Describe the movement of the cylinder and use the energy arguments to explain the movement.

h) Show that the vertical component of the spring force can be written as

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i) What is the normal force N acting on the cylinder from the rod when the spring is in its equilibrium length l_0?

j) Show that the normal force N(x) as a function of the position x from the rod on the cylinder is:

Also discuss the direction of the normal force.

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*Remember to attach all program code for each subtask*