# Velocity of current-carrying cylinder

• steve B. 98
In summary, a cylindrical rod of length w carrying a current I rolls without slipping on perpendicular rails while being bathed in a perpendicular field B. Its exit velocity, v, is given by the equation ##v = \sqrt{ \frac{4BLIw}{3M}}##, where M is the moment of inertia and L is the distance traveled. To solve for v, you must first find the net force on the rod, which is the force from the current and field minus the frictional force that causes rotation. Then, using the formula fr=Iα, solve for the net acceleration and plug it into the equation v^2=2aL. This should result in the correct answer, accounting for a factor of
steve B. 98

## Homework Statement

A cylindrical rod of length w carries a current I as shown(perpendicular to B) and is bathed in a field B perpendicular to the plane in which is a " shaped rail. The rod rolls without slipping on the rails, its length perpendicular to the two parallel rails and equal to the space between them. It starts at rest and rolls off after going a distance L. Show that its exit velocity is ##v = \sqrt{ \frac{4BLIw}{3M}}##

## Homework Equations

##F=B \times I w=BIw##
##\tau=r \times F##
##\tau=I\alpha##

## The Attempt at a Solution

[/B]
I start by taking the torque
##\tau=r \times BIw##
Using the moment of inertia for a cylinder I get
##\tau=Mr^2\alpha /2 ##
##\alpha=\frac{2BIw}{Mr}##
since the problem was rolling without slipping we get.
##a=\frac{2BIw}{M}##
Using ##v^2=0^2+2as##
##v=\sqrt{\frac{4BIwL}{M}}##
But this is off by a factor of ##\frac{1}{\sqrt{3}}##.
What did I do wrong, and some hints toward the solution would be nice.

easy question pal i have solved it and i am placing the pictures.
First find net force which is force on wire by current and field and second is frictional force which will rotate it and hence net force is IwB-f which is Ma .
Then find torque which is fr=Iα when you will sove net acc. to be 2IwB/(3M) place it in v^2=2aL you will get the answer.

#### Attachments

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## 1. What is the velocity of a current-carrying cylinder?

The velocity of a current-carrying cylinder refers to the speed at which the cylinder moves due to the flow of electric current through it.

## 2. How is the velocity of a current-carrying cylinder calculated?

The velocity of a current-carrying cylinder can be calculated using the equation v = I/(nAe), where I is the current, n is the number of electrons per unit volume, A is the cross-sectional area of the cylinder, and e is the charge of an electron.

## 3. What factors affect the velocity of a current-carrying cylinder?

The velocity of a current-carrying cylinder is affected by the strength of the current, the number of electrons in the cylinder, and the cross-sectional area of the cylinder.

## 4. How does the velocity of a current-carrying cylinder relate to Ohm's law?

According to Ohm's law, the velocity of a current-carrying cylinder is directly proportional to the strength of the current and inversely proportional to the cross-sectional area of the cylinder. This means that as the current increases, the velocity of the cylinder will also increase, and as the cross-sectional area decreases, the velocity will increase as well.

## 5. What is the significance of understanding the velocity of a current-carrying cylinder?

Understanding the velocity of a current-carrying cylinder is important in various fields such as electrical engineering and physics. It allows for the prediction and control of the movement of the cylinder, which is essential in designing and optimizing electrical systems and devices.

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