Stone tied to string whirled. Max string tension?

[itwazntme]

stone tied to string whirled. Max string tension??

Homework Statement

A stone of mass m at the end of a string of length l is whirled in a vertical circle at a constant speed when will the tension in the string be maximum?

Homework Equations

a) at the top of the circle
b) halfway down from the top
c) quarterway down from the top
d) at the bottom of the circle

The Attempt at a Solution

i know the answer, its obvious its the bottom of the circle due to force required to overcome the effect of gravity, but how can one prove it mathematically using the different physics phenomenon.

 

[Doc Al]

Analyze the forces acting on the stone at each of those points, then apply Newton's 2nd law. Hint: What's the acceleration of the stone at each point?

 

[kerimek]

I would approach this problem by using the equation for centripetal force, F=mv^2/r, where m is the mass of the stone, v is the velocity, and r is the radius of the circle.

At the top of the circle, the tension in the string would be equal to the weight of the stone, mg. This is because at the top of the circle, the centripetal force is equal to the weight of the stone, since there is no other force acting on the stone in the vertical direction.

Halfway down from the top, the tension in the string would be less than the weight of the stone, but greater than at the top. This is because at this point, the centripetal force is equal to the weight of the stone minus the tension in the string, which is acting in the opposite direction of the weight.

At a quarter of the way down from the top, the tension in the string would be even less, as the centripetal force is now equal to the weight of the stone minus the tension in the string, minus the weight of the stone again. This is because at this point, the weight of the stone is acting in the same direction as the tension in the string.

Finally, at the bottom of the circle, the tension in the string would be at its maximum. This is because at this point, the centripetal force is equal to the weight of the stone plus the tension in the string, as the weight of the stone is now acting in the opposite direction of the tension in the string.

Therefore, mathematically, we can prove that the tension in the string will be maximum at the bottom of the circle, as it is the point where the centripetal force is equal to the weight of the stone plus the tension in the string.