Finding the Least Speed for Rotating Eggs in a Basket

[alchemist]

some eggs are placed in a basket. the basket is safely rotated in a vertical plane so that the eggs describe a circle of radius 1 metre. what is the least speed at the top of the circle at which the eggs must move? the ans is 3.2m/s

in this question, i managed to determine the forces present...the weight, centripetal force, i know i am suppose to form 2 equations and solve it, but i have absolutely no idea how to do it...

i could only obtain one eqn: mv*2/r=mg-reaction (at the top of the rotation)
at the bottom it would simply be mv*2/r=reaction-mg

but how do you solve this and obtain the least speed?
are there any other eqn?

 

[HallsofIvy]

How do you "know" you are supposed to form 2 equations?

Actually your formula is correct. The point is that "least speed" at which the eggs will not fall is the one at which they become "weightless"- at which there is NO reaction.

The equation you need to solve is mv^2/r= mg.

 

[M98Ranger]

To find the least speed at which the eggs must move at the top of the circle, we need to consider the forces acting on the eggs. As mentioned, there are two main forces at play here: the weight of the eggs and the centripetal force required to keep them moving in a circular path.

The equation you have written, mv^2/r = mg - reaction, is correct. This equation represents the sum of the forces acting on the eggs at the top of the circle. However, we can rewrite it as mv^2/r + mg = reaction.

At the bottom of the circle, the eggs are moving with the same speed but the direction of the forces acting on them has changed. The centripetal force is now acting upwards, while the weight of the eggs and the reaction force are acting downwards. So the equation at the bottom becomes mv^2/r = reaction + mg.

Now, to find the least speed, we need to equate the two equations. This is because at the least speed, the reaction force at the top of the circle will be equal to the reaction force at the bottom of the circle. So we can write:

mv^2/r + mg = mv^2/r + mg
Simplifying, we get:
mv^2/r = mv^2/r

This means that the speed at the top and bottom of the circle must be the same for the reaction force to be equal. So the least speed at the top of the circle would be the same as the speed at the bottom, which is given to be 3.2 m/s. Therefore, the least speed at which the eggs must move at the top of the circle is also 3.2 m/s.

I hope this helps to clarify the concept and solve the problem. Remember, when dealing with circular motion, it is important to consider the forces involved and use equations that relate them to each other.