Review. The bar of mass m in Figure P30.51 is pulled horizontally across parallel, frictionless rails by a massless string that passes over a light, frictionless pulley and is attached to a suspended object of mass M. The uniform upward magnetic field has a magnitude B, and the distance between the rails is ℓ. The only significant electrical resistance is the load resistor R shown connecting the rails at one end. Assuming the suspended object is released with the bar at rest at t = 0, derive an expression that gives the bar’s horizontal speed as a function of time.
Figure P30.51
Answer to Problem 51CP
Explanation of Solution
Given info: Mass of bar is
The emf induced in the bar can be given as,
Here,
The current induced in the bar can be given as,
Here,
Substitute
The force induced in the bar due to magnetic field can be given as,
Here,
Substitute
The force due to weight can be given as,
Here,
As, force due to magnetic field and force due to weight will act in opposite direction, the net force acting on the bar can be given by subtracting equation (2) from equation (1),
Here,
The net force can also be given as,
Substitute
The equation (4) is a linear differential equation of the form,
Here,
The integrating factor for the equation (4) can be given as,
Here,
The solution for the differential equation is,
Here,
Substitute
Apply boundary condition,
Substitute
Thus, the expression for speed of the bar is
Conclusion:
Therefore, the expression for horizontal speed of the bar as a function of time will be
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Chapter 30 Solutions
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