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.  The rails are also connected to a voltage source V, so a current loop is formed.
The rod begins to move if the externally applied vertical magnetic field in which the rod is located reaches the value B. 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.

Part A

Find μ_s, 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

Concepts and reason

The main concepts used to solve the problem are ohm’s law, force due to magnetic field, and frictional force.

Initially, use the ohm’s law to find the current, Later, use the force due to magnetic field and use the current found in previous step to find the magnetic force. Finally, use the frictional force and equate the frictional force with the magnetic force to calculate the coefficient of static friction.

Fundamentals

The ohm’s law is expressed as follows:

Here, I is the current, R is the resistance, and V is the voltage.

The magnetic force on the rod due to applied voltage is expressed as follows:

Here, B is the magnetic field, I is the current, L is the length of the rod, and F is the force.

The frictional force is expressed as follow:

Here, is the static frictional force, is the coefficient of static friction, m is the mass, and g is the gravity.

(A)

Calculate the expression to calculate the static friction constant.

The ohm’s law is expressed as follows:

Here, I is the current, R is the resistance, and V is the voltage.

Rewrite the expression for current I.

Here, I is the current, R is the resistance, and V is the voltage.

The magnetic force on the rod due to applied voltage is expressed as follows:

Here, B is the magnetic field, I is the current, L is the length of the rod, and F is the force.

Substitute for I in expression .

The frictional force is expressed as follow:

Here, is the static frictional force, is the coefficient of static friction, m is the mass, and g is the gravity.

The current flows in the y-direction. So, conductor slides when magnetic field is B. this should be equal to frictional force.

Equate the frictional force and magnetic force and solve for coefficient of static friction.

Ans: Part A

Thus, the coefficient of static friction between the rod and the rails is .