I need this problem answered completely with complete steps. Please if you can\'

Question

I need this problem answered completely with complete steps. Please if you can't do this problem, donn't accept it. Thank you.

3. The figure below shows an axle whose wheels make good electrical contact with perfectly conducting rails set on a 30° inclined plane. The bottom ends of the rails are electrically connected by a perfect conductor. The system is in a uniform magnetic field B directed upward. If the axle is allowed to roll down the inclined plane, it does not accelerate uniformly. Instead, it soon attains a constant terminal velocity vt and then continues to descend at that velocity. Derive an expression for vr in terms of B, R, a, and m, where B is the magnitude of the vertical magnetic field, R is the electrical resistance of the axle, a is its length, and m is its mass. 300

Explanation / Answer

Since the axle was moving with constant velocity ,hence accelaraion is zero so net force is zero

which means that downward Force = upward force along the incline

m*g*sin(30) = i*a*B*cos(30)

i is the current in the axle

i = e/R

e is the motional emf = B*cos(theta)*a*Vt

then

m*g*sin(30) = i*a*B*cos(30)

m*g*tan(30) = (B*cos(30)*a*Vt/R)*a*B

m*9.8*tan(30)/cos(30) = B^2*a^2*Vt/R

6.53*m = B^2*a^2*Vt/R

therefore

Vt = 6.53*m*R/(B*a)^2

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