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The situation :

A block of mass $M$ is tied to a string and is spun around in a vertical circle.

The question asks me to calculate the tension in the string 'at the lowest point' after giving some values.

In the problem, when the body is in the lowest point, shouldn't the tension be $Mg$ only?

It is told that perpendicular forces are independent of each other, then why/why not the velocity of the mass be included in this scope of calculating the tension in the wire?

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To keep the mass going around in a circle, you need to be accelerating towards the center of the circle. The force for this (if your speed is $v$) is

$$F = \frac{mv^2}{r}$$

where $m$ is the mass, and $r$ is the radius of the circle.

Now $v$ will be a function of position in your arc - you will have to figure out what it is at the bottom (conservation of energy may be your friend).

To get the tension in the string, this force has to be added to the force of gravity at the bottom (and it is subtracted at the top because gravity will be pointing in the same direction as the force needed to keep the mass in orbit - so you need less tension in the string).

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If the mass is moving in a circle then forces have to act on it in order to keep its track. Otherwise it will move in a straight line. And with gravity $Mg$ alone it will move in a parabola. So there is more to it to get it going in a circle.

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