How is the induced EMF calculated in a circuit with a swinging pendulum gate?

[IHateMayonnaise]

Homework Statement

I just took a test, and I am very unsure of the validity of how I approached the problem. Just looking for some feedback cause this is bugging me!

There is a circuit in the xz-plane (vertical), and the circuit has on it a gate. The gate is a pendulum of length L, which swings back and forth at some velocity (given), and when it is straight up and down (theta=0) it makes contact with the other lead and completes the circuit. All the time there is a CONSTANT magnetic field B pointing normal to the circuit (+y). What is the induced EMF?

The velocity as a function of time is given:

$$\dot{x}=\omega x_o \cos\left(\omega t\right)$$

Homework Equations

$$\mathcal{E}=-\frac{1}{c} \frac{d \Phi}{dt}$$
$$\Phi = \oint_S \mathbf{B}\cdot\mathbf{dS}$$

The Attempt at a Solution

The initial conditions can be deduced fairly easily ($x_o$ is the horizontal amplitude of the pendulum): $\dot{x}(0)=0, x(0)=x_o$. From the initial conditions and integrating,

$$x(t) = x_o\left[\sin\left(\frac{n\pi t}{2\tau}\right)+1\right]$$

My (probably flawed) methodology is what follows. I thought it easier to think of the circuit as staying constant (eg no gate) and the field is pulsing at some rate, which I assume to directly correspond to that of the pendulum. From this we can then calculate the induced emf. Basically from here, all I did was take the above equation, and instead of the position amplitude $x_o$ I substituted the field amplitude $B_o$.

$$B(t) = B_o\left[\sin\left(\frac{n\pi t}{2\tau}\right)+1\right]$$

From here I just took the derivative and said I was done, but I feel so, so dirty and I know I goofed this up. Thoughts? Thanks yall

IHateMayonnaise

 

[kreil]

"The magnetic flux is a measure of the number of magnetic field lines passing through an area...If the flux changes, an emf will be induced. There are therefore three ways an emf can be induced in a loop:

1. Change the magnetic field
2. Change the area of the loop
3. Change the angle between the field and the loop"

Which one applies here?More here:
http://physics.bu.edu/~duffy/py106/InducedEMF.html