# Simple pendulum period in three different cases

Imagine you have a simple pendulum hanging on the ceiling of a train which has a period called T. How will the period be in the following cases:

1. When the train is in circular motion in a curve of radius R with constant speed (I think it would be lower than T (the initial period), that's my intuition).
2. The train is going up a constant slope with constant speed (I also think it would be lower than T).
3. The train moves over a hill of radius R with constant speed. (I feel it would be greater than T because the pendulum just moves in step with the change of the slope. That's why the pendulum oscillates completely just when the train has left the hill).

Sorry for my poor explanations. I have just tried to figure it out the best way I could. What do you think? Do you agree with me?

Thanks

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The period of the pendulum is roughly

$T\approx2\pi\sqrt{\frac{L}{F}}$

Where $L$ is its length and $F$ is the downward-pulling force (usually gravity). Let's call our reference period $T_{rest}$.

1. When the train is in circular motion in a curve of radius R with constant speed.

If the train is in circular motion the pendulum experiences a fictitious centrifugal force that points radially outward. Firstly, that means that the total force on the pendulum is not straight down any more, but somewhat diagonally downward/outward. Since you have to add this force (vectorially) to the gravitational force, $F$ will also be larger than before. Looking at our equation above, this means that indeed $T_1 < T_{rest}$.

2. The train is going up a constant slope with constant speed.

If the train travels along a straight line (and a constant slope is a straight line) at constant speed, then the pendulum is in an inertial frame. Therefore, there are no fictitious or other forces adding to gravity. Hence, $T_2 = T_{rest}$.

3. The train moves over a hill of radius R with constant speed.

This one is tougher. The centrifugal force is still pointing radially outwards - but now it is not perpendicular to the gravitational force any more. For instance, at the top of the hill, it is pointing upward, and thus cancelling gravity partly. At other positions it will point diagonally upward. This case needs its own somewhat more involved treatment, because the total $F$ is not constant. Therefore, $T_3$ will also change gradually. In particular, it will be maximal at the top of the hill, and will decrease symmetrically down both sides of the hill.

Sorry that I don't have any pictures to clarify this. Feel free to ask if anything is unclear.

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