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A metal crossbar of mass M and length L slides with negligible friction but good

ID: 1362665 • Letter: A

Question

A metal crossbar of mass M and length L slides with negligible friction but good electrical contact down between two very long vertical metal rails (not drawn to scale). The crossbar and rails have negligible electrical resistance, but the rails are connected at the bottom by a wire with resistance R. Throughout the entire region is a uniform magnetic field B pointing directly out of the page. If released from rest, the bar's downward acceleration will start at gbut decrease as it asymptotically approaches a constant speed of (called its "terminal velocity"). Explain why the bar approaches a constant velocity rather than maintaining a constant acceleration of g, even if air drag is negligible. Find an expression for the terminal speed v_T in terms of the given properties of the system. Demonstrate that the electrical power dissipated in the resistor is equal to the rate at which the Earth-crossbar system loses gravitational potential energy, for any crossbar speed.

Explanation / Answer

a)

here ,as the velocity of the rod is increased ,

the induced current in the rod will increase

and hence , the magnetic force will increase with induced current

after some time , magnetic force will be equal to gravity force

Now , this velocity will stay constant

b)

for the induced current

I = B * vt * L/R

Now for terminal velocity

B*I * L = M * g

(B * vt * L/R) * I * L = M * g

Vt = R * M * g/(B^2 * L^2 )

the terminal velocity is R * M * g/(B^2 * L^2 )

c)

for any velocity , v

rate of loss of gravitational potential energy = Force * velocity

rate of loss of gravitational potential energy = M* g * v

Now ,for the energy dissiated in resistor

I = B * v * L/R

energy = I^2 * R

energy = B^2 * v^2 * L^2/R

as B*I*L = m * g ( for constant speed)

energy = m * g* v

the rate of disspation of resitor heat is equal to rate of loss of gravitational potential energy