# Tension in a Pendulum String / Rod

I'm making a pendulum simulation, so far everything works! I have a free moving pendulum following simple harmonic motion that hangs from a point. My problem is that I would like the pendulum to act appropriately when the point it hangs from is moved (in x and/or y directions). To do this, my approach is to calculate the tension in the light inextensible string / rod. I can then calculate the angular acceleration caused by this.

When the fixed point is not moving, I said the following:
From SHM:
centripetal force = mv^2 / l

where m is mass, v is velocity (I'm using v = angular velocity x l) and l is the length of the pendulum string.

The force exerted from the mass is mgcos theta. Where g is gravity and theta is measured from the vertical. I said this from the thinking that if the pendulum is stationary directly under the point it's hanging from then the force on the rod from weight must be mg, and if it's at 90 degrees, the force must be 0.

In relation to the swinging mass: T acts upward, centripetal force acts towards the centre, ie upward and mgcos theta acts downward.

That gives us: T = mgcos theta - mv^2 / l.

Am I right?

Now, the point from which the pendulum is suspended can be moved. Moving it this will increase or decrease the tension. So I assume I will simply add another term onto the equation for T. The force exerted must be proportional to the displacement of the point.

This is where I get iffy though, I'm not really sure how the tension will be affected by the force. Nor am I sure of what the force applied to the top point actually is.

Any help would be great, thanks!