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 m and electrical resistance R rests on parallel horizontal rails (that have negligible electric resistance), which are a distance L apart, as shown in (Figure 1). The rails are also connected to a voltage source V, so a current loop is formed.

The vertical magnetic field, initially zero, is slowly increased. When the field strength reaches the value B0, the rod, which was initially at rest, begins to move. Assume that the rod has a slightly flattened bottom so that it slides instead of rolling. Use g for the magnitude of the acceleration due to gravity.

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

Express the coefficient of static friction in terms of variables given in the introduction.

Answer 1

The force on the metal rod due to current in the magnetic field is

The current in the loop is

That gives us

The mass of the metal rod is m.

So the force of static friction on the metal rod is

Now, we know the bar starts to move when and only when the magnetic field reaches a minimum value of Bo. So, it is clear that the magnetic force at this value just cancels out the force due to friction. i.e

So, the coefficient of static friction is

Where,

is the coefficient of static friction.

is the minimum magnetic field to start the rod moving.

is the distance between the rails.

is the potential difference between the rails.

is the mass of the metal rod.

is acceleration due to gravity.

is the electrical resistance.