A conducting solid cylinder of mass \"m\", length \"L\", resistance \"R\" and ra

Question

A conducting solid cylinder of mass "m", length "L", resistance "R" and radius "r" is placed on the rails at t = 0 as shown. The rails are inclined 30 degrees from the horizontal and the cylinder starts at an initial position "x_0" from the upper end. A batten of voltaic "V" is connected to the upper end of the rails and the lower end is open. The rails have no resistance. There is a constant magnetic field "B" projected vertically upward through the rails. When the cylinder is placed on the rails the circuit is closed causing a current in the circuit. The static friction between the cylinder and rails is large enough so that no slippage occurs. Given [m, L, R, r, V, B| Determine: The magnetic flux as a function of position. The EMF as a function of velocity. The current as a function of velocity. The magnetic force as a function of velocity. The velocity of the cylinder as a function of time. The velocity of the cylinder after a long time (terminal velocity). The kinetic energy (translational and rotational) as a function of time. The power output of the battery as a function of time. The power lost to heat as a function of time. Prove that the power given to the circuit from the rate of change in gravitational potential energy added to the power produced by the battery is equal to the rate of change in the kinetic energy added to the power lost to heat. After a long time, the cylinder runs off the rails and onto a horizontal set of rails. The new set of rails has no battery connected to it and a wire with no resistance ties the far end of the rails together. Again the static friction is large enough so that no slippage occurs. Reset the position and time to zero at this point. Determine: The velocity of the cylinder as a function of time. The distance the cylinder rolls before coming to rest, m. The current as a function of time. Prove that on this horizontal section, the original kinetic energy is equal to the heat dissipated by the resistor.

Explanation / Answer

a) magnetic flux = BA cos 30 degree

                         = BLx cos 30 degree where x is its position from battery along the incline.

b) emf = dflux/dt = BL cos 30 degree* dx/dt

                 = BLv cos 30 degree

c) current i = [V-emf]/R

             = [V-BLvcos 30 degree]/R

d) Force = iLB

            =   LB*[V-BLvcos 30 degree]/R

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.