Friday, January 3, 2014

A physics problem and test of LaTeX

Let's look at a problem accessible to students taking their first course in physics.

An object is on a horizontal table, 1m from the edge. It is pushed so that it travels the 1m in 2s, then falls off the edge of the table. Is it likely that the object was on wheels?

Reasoning: Let's imagine giving the object a push and it slides (no wheels). Let's compute the maximum coefficient of kinetic friction using the simple model of friction we learned. Let's also neglect another type of friction, air resistance (which would be much smaller than sliding friction.) Consistent with the problem description, the maximum friction would occur if the object just makes it to the edge before it falls off the edge. Mathematically we can say that the final speed is zero, or \( v_{f}=0\). We can then compute the initial speed to be $$v_{avg} = \frac{1}{2}(v_{i} + v_{f}) = \frac{\Delta x}{\Delta t}$$ so, using \( v_{f}=0\) and \( \Delta x=1m\) and \( \Delta t=2s\), $$v_{i}=2\frac{1m}{2s}=1m/s.$$ We can then get the acceleration: $$a=\frac{v_{f}-v_{i}}{\Delta t}=\frac{-1m/s}{2s}=-\frac{1}{2}m/s^{2}$$ This acceleration, by assumption, is caused by kinetic friction. Here we have the simple case where the normal force is equal to the weight, so from \( F = -\mu_{k} N = -\mu_{k}mg = ma\) we get: $$\mu_{k} = -\frac{a}{g} = \frac{0.5}{9.8} = .051$$ Looking at a table of coefficients of friction, we see that this is a very small value, in the range of lubricated metal or ice on ice. So it is a reasonable guess that the friction is so low that the object was rolling, not sliding.

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