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Solution
a) What is the horizontal position 
of the cylinder when the spring is at its natural equilibrium length of
= 0.5 m? Define the attachment
point of the spring on the table as position 𝑥 = zero.
b. What is the length 𝑙
of the spring when the cylinder is at any given position 𝑥?
Solution
a) Draw a free-body diagram of the cylinder
d) Show that the horizontal force, which the spring exerts on the cylinder, is:
e) Plot the horizontal spring force as a function of position of the cylinder between 𝑥 = − 0.75 m and 𝑥 = +0.75 m and explain what you expect for the motion of the cylinder
Graph:
to the position 𝑥
= 0.6 m and let it go without giving it any initial velocity. Write a program
to solve the equation of motion numerically and plot the position and velocity
of the cylinder for the first 10 seconds of its motion.
Solution
Equations of the motions are given below;
To solve the equation
of the motion numerically is shown as above; and the plot of the position and
the initial velocity in the MATLAB is done by using the ODE 45 as shown in
MATALB code.
Graph
g) Run your program again, but this time
choose a starting position of 𝑥
= 0.65 m. Plot the position and velocity as a function of time for the first 10
seconds of motion.
Equations of the motions are given below;
Graph:
h) Show that the vertical component of the spring force acting on the cylinder is:
i. What is the normal force from the rod acting on the cylinder when the spring is at its natural equilibrium length?
j) What is the normal force 𝑁
from the rod acting on the cylinder when the cylinder is at a position 𝑥?
Solution
k) Show that the normal force becomes zero for:
l)
We will now consider a more realistic model and include friction. Due to
polished surfaces and the use of a lubricant the (dynamic) friction coefficient
is only 𝜇
= 0.05. Modify your program to include friction. Start at the position 𝑥 = 0.75 m and plot again position and
velocity as a function of time.
Solution
By using the
MATALB as explained in the part f & g, but in this part we consider the realistic
model and include friction, so the friction coefficients is 𝜇 = 0.05, and in
this part the position 𝑥
= 0.75 m.
MATLAB Code:
m) Make an additional figure where you plot the kinetic energy versus
the position for the first 10 seconds of the motion. Explain the results.
Solution
Position 0: 0.01:0.75
MATLAB Code
As shown the above formula of kinetic energy; our requirements is to plot the position and the kinetic energy, first of all we find the initial velocity as shown in above calculations by using the position 0.75 then by using the 1st equation of motion and calculate the final velocity. AT last by using the MATLAB plot the kinetic energy and positions
Result
n) Find the work that is needed to bring the cylinder from its
equilibrium position (that you have found in part a) to the position 𝑥 = 0.75 m numerically.
Solution
o) Write a program to find the potential energy for the spring force as
a function of the position of the cylinder. Plot the result and compare with
the kinetic energy from part m). You can use the function “cumtrapz” or program
your own function for the numerical integration.
Solution
MATLAB Code
By using the spring
constant k= 500; and the position as the function of the spring forces that
found in the part a, and the 0.75
Position = 0.75; 20.9848
Now the part m, the kinetic energy and the part O, the
potential energy compare with each, you can see that in the graph, the part m
has the .Exponential curve, and the potential energy has the linear curve.
p) Where are the equilibrium points of the
system? Characterize them as stable or unstable.
Solution
The System with the equilibrium points;
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