A wire of length L and resistance R forms a rectangular loop twice as long as it

Question

A wire of length L and resistance R forms a rectangular loop twice as long as it is wide. It's mounted on a nonconducting horizontal axle parallel to its longer dimension, as shown in the figure. A uniform magnetic field B points into the page. A long string of negligible mass is wrapped many times around a drum of radius a attached to the axle, and a mass m is attached to the string. When the mass is released, it falls and eventually reaches a speed that, averaged over one cycle of the loop's rotation, is constant Find an expression for that average speed.

Explanation / Answer

torque due to hanging mass,

torque = a m g

so torque on coil be of same magnitude but in opposite direction .

emf induced in coil = N A B w

current = I = N A B w / R

torque = (W ) (i l B)

= wl N A B^2 w / R = N A^2 B^2 w / R

and l + w = L

2w + w = L => w = L/3

A = w l = L^2 / 3

1 x L^4 /9 x B^2 w / R = m g a

w = 9 R m g a / (L^2 B^2 )

and w = v / a = 9 R m g / (L^2 B^2 )

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