# Moving a Pendulum's Hanigng Point and the Tension in the String

I worked out the tension in a pendulum string to be T = mgcos theta + mv^2 / r where r is the length of the string, m is the mass of the bob, v is the bob's velocity, g is acceleration due to gravity and theta is measured from the vertical.

I got this equation from the forces acting on the string. Centripetal force is mv^2 / r and the weight of the bob acting on the string will be mgcos theta.

But what if the point the pendulum string is attached to is moved a distance dx in the x-direction and dy in the y-direction? Obviously, the tension in the string will change.

What's the new formula for T? I just can't work it out!

I observed a real pendulum and found, expectedly, that if you move the hanging point to the left, the bob will swing right in relation to the hanging point - the opposite is true for moving the hanging point left. However, I found that if if I move the hanging point up, the bob will accelerate in the direction it's already travelling. When moving the hanging point down, I observed both accelerations and decelerations and could not figure out exactly why this was. What's going on?

Thanks a lot!

EDIT:
Using SUVAT: s = ut + 0.5at^2
Rearranging: a = 2(s - ut) / t^2

Now, say that s = dx for the x-direction and s = dy for the y-direction.

As F = ma, we can work out the extra (or lesser) Tension in the x and y directions. Although, I'm a little confused as to what mass we would use. The mass of the whole system?

Is this correct? In a simulation it provides some slightly odd results in certain situations, and has to be scaled down drastically!

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