Please this extra credit will help me pass the class, please show me how to do o

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Please this extra credit will help me pass the class, please show me how to do only [ i,j, k,l,m,n] I GOT THE FIRST 8 QUESTIONS. Please I need someone helps

(10 Points) A conducting solid cylinder of mass 'm length , resistance "R and radius “ "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 initia position "x," from the upper end. A battery of voltage "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 as the cylinder rolls down. Given [m, L, R, r, V, B, C, x] Determine a. The magnetic flux through the circuit as a function of position. b. The induced EMF as a function of velocity. c. The current in the circuit as a function of velocity d. The magnetic force on the cylinder as a function of velocity e. The velocity of the cylinder center of mass as a function of time. r. The velocity of the cylinder after a long time (terminal velocity). g. The kinetic energy (translational and rotational) as a function of time. h. The power output of the battery as a function of time i. The power lost to heat as a function of time. j. 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 (10 Points) After a long time, the cylinder runs off the rails and onto another 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. Reset the position and time to zero at this point. Determinc: k. The velocity of the cylinder as a function of time. I. The distance the cylinder rolls before coming to rest. m. The current as a function of time. n. Prove that the original kinetic energy is equal to the heat dissipated by the resistor.

Explanation / Answer

given mass = m

length = L

resistance = R

radius = r

theta = 30 deg

initial position = xo

voltage= V

magnetic field = B ( upwards)

a. let the position be at x

then magnetic flux = B*cos(theta)*x*l = Bxl*cos(theta)

b. induced EMF = -d(phi)/dt = -Blcos(theta)*v

c. current in the circuit = i

V - Blcos(theta)*v = iR

i = (V - Blcos(theta)*v)/R

d. magnetic force on the cylinder = F = Bil = B(V - Blcos(theta)*v)l/R

e. now

Fb - mgsin(theta) - f = ma

and

f*r = 0.5mr^2*alpha

but for rolling without slipping

alpha = a/r

hence

f = 0.5ma

hence

Fb - mgsin(theta) = 1.5ma

B(V - Blcos(theta)*v)l/R - mgsin(theta) = 1.5dv/dt

1.5dv/B(V - Blcos(theta)*v)l/R - mgsin(theta) = dt

integrating form v = 0 to v = v

1.5R*ln(B(V - Blcos(theta)*v)l/R - mgsin(theta) ) = -B^2*lcos(theta)l*t

(V - (R/Bl)(mgsin(theta) + e^(-B^2*lcos(theta)l*t/1.5R)) = Blcos(theta)*v

v = (V - (R/Bl)(mgsin(theta) + e^(-B^2*lcos(theta)l*t/1.5R))/Blcos(theta)

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