# Solve Homework Problem: Velocity & Force of Rod in Rails

• erisedk
In summary, the problem involves a free sliding rod on parallel rails with a magnetic field and a variable external force applied to maintain a constant current. The aim is to find the velocity of the rod and the applied force as a function of the distance of the rod from a specific point. The solution involves using the formula F=mv(dv/dx) and setting the net force equal to 0, leading to the expression v = c(R+2λx), where c is an arbitrary constant. The unknown variables, including the initial velocity and position, need to be specified in order to eliminate the constant of integration.
erisedk

## Homework Statement

Two long parallel horizontal rails, a distance d apart and each having a resistance λ per unit length, are joined at one end by a resistance R. A perfectly conducting rod MN of mass m is free to slide along the rails without friction (see the figure). There is a uniform magnetic field of induction B normal to the plane of the paper directed into the paper. A variable force F is applied to the rod MN such that, as the rod moves, a constant current flows through R.
Find the velocity of the rod and the applied force F as a function of the distance x of the rod from R.
http://www.zigya.com/application/zrc/images/qvar/PHEN12050806.png

## The Attempt at a Solution

The emf induced will be Bvd, where v is the velocity of the rod at that instant.
The current is i = emf/resistance = Bvd/(R+2λx).
I know F = FB = idB = B2d2v/(R+λx)
However, I don't know how to get rid of v in the F expression, and I don't know how to express v in terms of x. So, my aim is to figure out v, because then I'll get F as well.
I figured since the current is constant, di/dx = 0
##\frac{d}{dx} \frac{Bvd}{R+2λx}## = 0

##( \frac{dv}{dx}) (R+2λx) = (v)(2λ)##

## \int \frac{1}{v} \, dv = \int \frac{2λ}{R+2λx} \, dx ##

## v = c(R+2λx) ## where c is an arbitrary constant.

erisedk said:
F = B2d2v/(R+λx)
You have mass 'm' of the rod. You can replace F by m(dv/dt) and further using chain rule, F=mv(dv/dx).

cnh1995 said:
You have mass 'm' of the rod. You can replace F by m(dv/dt) and further using chain rule, F=mv(dv/dx).

This might be a really dumb question but isn't Fnet = ma? And here Fnet = 0 as the variable force F is equal and opposite to the Lorentz force at any instant? I got the answer with this approach though. I just have this stupid question.

erisedk said:
This might be a really dumb question but isn't Fnet = ma? And here Fnet = 0 as the variable force F is equal and opposite to the Lorentz force at any instant? I got the answer with this approach though. I just have this stupid question.
If the net force were 0, the rod would move with a constant velocity and the current would be decreasing since the resistance of the loop is increasing with distance.

Instead of what I said in #2, I believe your approach in the OP is correct.
Here's how I think the problem should be..The rod is initially moving with some velocity u at some distance xo from the starting point, which drives a current Bud/(R+2λxo). A variable force is applied to maintain this current. I believe u and xo need to be specified. If xo is assumed to be 0, still u is unknown. If xo and u are known, constant of integration in your solution attempt can be eliminated.

cnh1995 said:
If the net force were 0, the rod would move with a constant velocity and the current would be decreasing since the resistance of the loop is increasing with distance.

erisedk said:
F = FB = idB = B2d2v/(R+λx)
Wouldn't this be incorrect then? Because net force would be F - FB, I guess. I wrote F (variable external force) = FB (Lorentz force) assuming that they would be equal and opposite. However, I do agree with your conclusion that the current would then be decreasing and we'd have a constant velocity. I'm just not sure how to resolve F = FB dilemma.

## 1. What is the formula for calculating velocity?

The formula for calculating velocity is velocity = distance/time. This means that velocity is equal to the distance traveled divided by the time it took to travel that distance.

## 2. How do you calculate force?

The formula for calculating force is force = mass x acceleration. This means that force is equal to the mass of an object multiplied by its acceleration.

## 3. How do you solve a homework problem involving velocity and force?

To solve a homework problem involving velocity and force, first identify the given information and what is being asked for. Then, use the appropriate formulas to calculate the missing values. Be sure to pay attention to units and use the correct formula for the given scenario.

## 4. What are some common units of measurement for velocity and force?

The most common units of measurement for velocity are meters per second (m/s) or miles per hour (mph). The most common units of measurement for force are newtons (N) or pounds (lb).

## 5. Can you provide an example problem involving velocity and force?

Sure, here is an example problem: A 2 kg ball is rolling down a ramp with an acceleration of 5 m/s^2. What is the force exerted on the ball? To solve this, we can use the formula force = mass x acceleration. Plugging in the values, we get force = 2 kg x 5 m/s^2 = 10 N. Therefore, the force exerted on the ball is 10 newtons.

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