In this project we address the motion of an object sliding on a slippery
surface-such as a ski sliding in a snowy track. You will learn how to find the
equation of motion for sliding systems both analytically and numerically, and
to interpret the results.
We start by studying a simplified situation called frictional motion: A block
is sliding on a surface. moving with a velocity \(v\) in the positive
\(x\)-direction. The forces from the interactions with the surface results in an
acceleration:
$$
a=\left\\{\begin{array}{cc}
-\mu(|v|) g & v>0 \\
0 & v=0 \\
\mu(|v|) g & v<0
\end{array}\right.
$$
where \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) is the acceleration of gravity. Let
us first assume that \(\mu(v)=\) \(\mu=0.1\) for the surface. That is, we assume
that the coefficient of friction does not depend on the velocity of the block.
We give the block a push and release it with a velocity of \(5 \mathrm{~m} /
\mathrm{s}\). (a) Find the velocity, \(v(t)\), of the block.
(b) How long time does it take until the block stops?
(c) Write a program where you find \(v(t)\) numerically using Euler's or Euler-
Cromer's method. (Hint: You can find a program example in the textbook.) Use
the program to plot \(v(t)\) and compare with your analytical solution. Use a
timestep of \(\Delta t=0.01 .\)
The description of friction provided above is too simplified. The coefficient
of friction is generally not independent of velocity. For dry friction, the
coefficient of friction can in some cases be approximated by the following
formula:
$$
\mu(v)=\mu_{d}+\frac{\mu_{s}-\mu_{d}}{1+v / v^{*}}
$$
where \(\mu_{d}=0.1\) often is called the dynamic coefficient of friction,
\(\mu_{s}=0.2\) is called the static coefficient of friction, and \(v^{*}=0.5
\mathrm{~m} / \mathrm{s}\) is a characteristic velocity for the contact between
the block and the surface.
(d) Show that the acceleration of the block is:
$$
a(v)=-\mu_{d g}-g \frac{\mu_{s}-\mu_{d}}{1+v / v^{*}}
$$
for \(v>0\).
(e) Use your program to find \(v(t)\) for the more realistic model, with the
same starting velocity, and compare with your previous results. Are your
results reasonable? Explain.
The model we have presented so far is only relevant at small velocities. At
higher velocities the snow or ice melts, and the coefficient of friction
displays a different dependency on velocity:
$$
\mu(v)=\mu_{m}\left(\frac{v}{v_{m}}\right)^{-\frac{1}{2}} \text { when }
v>v_{m}
$$
where \(v_{m}\) is the velocity where melting becomes important. For lower
velocities the model presented above with static and dynamic friction is still
valid.
(f) Show that
$$
\mu_{m}=\mu_{d}+\frac{\mu_{s}-\mu_{d}}{1+v_{m} / v^{*}},
$$
in order for the coefficient of friction to be continuous at \(v=v_{m}\).
(g) Modify your program to find the time development of \(v\) for the block when
\(v_{m}=1.5 \mathrm{~m} / \mathrm{s}\). Compare with the two other models above:
The model without velocity dependence and the model for dry friction. Comment
on the results.
(h) The process may be clearer if you plot the acceleration for all the three
models in the same plot. Modify your program to plot \(a(t)\), plot the results,
and comment on the results. What would happen if the initial velocity was much
higher or much lower than \(5 \mathrm{~m} / \mathrm{s}\) ?
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