Question

A rail gun uses electromagnetic forces to accelerate a projectile to very high velocities. The basic mechanism of acceleration is relatively simple and can be illustrated in the following example. A metal rod of mass 50.0 g and electrical resistance 0.300 \Omega rests on parallel horizontal rails that have negligible electric resistance. The rails are a distance L = 7.00 cm apart. The rails are also connected to a voltage source providing a voltage of V = 5.00 V.

The rod is placed in a vertical magnetic field. The rod begins to slide when the field reaches the value B = 0.210 T. Assume that the rod has a slightly flattened bottom so that it slides instead of rolling. Use 9.80 m/s^2 for the magnitude of the acceleration due to gravity.

Find mu_s, the coefficient of static friction between the rod and the rails.

Answer 1

Concepts and reason

The concepts used to solve this problem are frictional force and magnetic force. The frictional force will act between the two surfaces. If the object is placed in the magnetic field. The magnetic force will act on this object. When the rod starts to move, then the applied force on the rod is equal to the static force of friction. Equate the expression of the frictional force and the magnetic force to get the coefficient of static friction. It is done according to the given condition in the problem. To find the value of the current, use Ohm's law.

Fundamentals

Write the expression of the frictional force. \(f=\mu m g\)

Here, \(\mu\) is the coefficient of static friction, \(m\) is the mass of the object and \(g\) is the acceleration due to gravity. Write the expression of the magnetic force. \(F=B I L\)

Here, \(B\) is the magnetic field, \(I\) is the current and \(L\) is the length. Write the mathematical expression of Ohm's law. \(V=I R\)

Here, \(I\) is current and \(R\) is the resistance.

Write the expression of the frictional force. \(f=\mu m g\)

Write the expression of the magnetic force. \(F=B I L\)

Equate the frictional force and magnetic force. \(\mu m g=B I L\)

Rearrange the expression to get the value of the coefficient of static friction. \(\mu=\frac{B I L}{m g} \ldots \ldots\) (1)

Explanation | Common mistakes | Hint for next step

In the given problem, the railgun uses electromagnetic force to accelerate a projectile. A metal rod rests on the parallel horizontal rails. The rails are at some distance. The rails are also connected to a voltage. The rods are placed in a vertical magnetic field. At a certain value of the magnetic field, the rod begins to slide. When the rod starts to move, then the applied force on the rod is equal to the static force of friction. Then, there will be a frictional force between the rod and the rails. Equate the frictional force and magnetic force to get the value of the coefficient of static friction between the rod and the rails.

Write the mathematical expression of Ohm's law. \(V=I R\)

Rearrange the expression to get the value of the current. \(I=\frac{V}{R}\)

Substitute \(\frac{V}{R}\) for \(I\) in equation (1). \(\mu=\frac{B\left(\frac{V}{R}\right) L}{m g}\)

Again, substitute \(0.210\) T for \(B, 7.00\) cm for \(L, 50.0\) g for \(m, 5.00 \mathrm{~V}\) for \(V, 0.300 \Omega\) for \(R\) and \(9.8 \mathrm{~m} \cdot \mathrm{s}^{-2}\).

$$ \begin{array}{c} \mu=\frac{(0.210 \mathrm{~T})\left(\frac{5.00 \mathrm{~V}}{0.300 \Omega}\right)(7.00 \mathrm{~cm})}{(50.0 \mathrm{~g})\left(9.8 \mathrm{~m} \cdot \mathrm{s}^{-2}\right)} \\ =\frac{(0.210 \mathrm{~T})\left(\frac{5.00 \mathrm{~V}}{0.300 \Omega}\right)(7.00 \mathrm{~cm})\left(\frac{1 \mathrm{~m}}{100 \mathrm{~cm}}\right)}{(50.0 \mathrm{~g})\left(\frac{1 \mathrm{~kg}}{1000 \mathrm{~g}}\right)\left(9.8 \mathrm{~m} \cdot \mathrm{s}^{-2}\right)} \\ =0.5 \end{array} $$

Use the mathematical expression of Ohm's law to calculate the current through the metal rod. Substitute the current in equation (1) which is calculated in step 1 . Then, substitute the value of the magnetic field, the length, mass, resistance, and the voltage to get the value of the coefficient of static friction.

The coefficient of static friction between the rod and the rails is \(0.5\).