A sphere of $0.5\,kg$ of mass is rotating around an axis in a vertical plane as shown in the picture from below. The minimum speed and the maximum speed of the sphere are $2\,\frac{m}{s}$ and $4\frac{m}{s}$ respectively. Find the maximum tension (measured in Newtons) that the wire can hold. Assume $g=10\,\frac{m}{s^2}$.

The alternatives are as follows:

$\begin{array}{ll}

1.&10\,N\\

2.&25\,N\\

3.&20\,N\\

4.&30\,N\\

\end{array}$

I'm confused exactly how to assess this problem. In order to find the maximum tension I believe the equation to get this is the conservation of mechanical energy.

My instinct tells me that the maximum tension will be on the bottom as follows:

Therefore:

At that point the forces acting will be the tension of the wire and the weight.

$T-mg=\frac{mv^2}{r}$

$T= \frac{mv^2}{r} + mg$

What's the speed at this point?. Should i assume $4\,frac{m}{s}$ or $2\frac{m}{s}$ and why?. what's the physical reason for it?.

I felt that (based on experience) that the speed will be at that point maximum. Hence I'll use $4\frac{m}{s}$.

Then the tension will be:

$T= \frac{0.5(4)^2}{r} + 0.5\times 10$

But the problem lies on what r should I use?.

Then I think this might come from the conservation of mechanical energy?.

$E_k+E_u=E'_k+E'_u$

$\textrm{E=Energy at the top}$

$\textrm{E'=Energy at the bottom}$

Now for this I'm assuming that the reference for establishing the height of the sphere is passing through the center or axis of rotation.

$\frac{1}{2}mv^2+mgr=\frac{1}{2}mu^2+mg(-r)$

$\frac{1}{2}(0.5)(2)^2+0.5(10)r=\frac{1}{2}(0.5)(4^2)+(0.5)(10)(-r)$

$r=0.3$

Therefore the radius is $0.3$

And the maximum tension of the wire will be: By replacing the earlier value in the above equation,

$T= \frac{0.5(4)^2}{0.3} + 0.5\times 10$

$T=31.66\,N$

However this answer does not appear within the alternatives.

But If I were to use this formula for the speed at the lowest point=

$v_2=\sqrt{5rg}$ (

**How was this formula derived?**)

Then $r= \frac{v^2_2}{5g}=\frac{4^2}{5*10}=\frac{8}{25}$

Introducing this in the earlier equation becomes into:

$T= \frac{0.5(4)^2}{\frac{8}{25}} + 0.5\times 10 = 30\,N\\$

Which does appear in one of the alternatives.

But If I were to use this formula instead:

$v_1=\sqrt{rg}$ (

**How was this formula derived?**)

$r=\frac{v^2_1}{g}=\frac{4}{10}=\frac{2}{5}$

Then the radius is different? Why is it so?.

Replacing this value in the earlier equation:

$T= \frac{0.5(4)^2}{\frac{2}{5}} + 0.5\times 10 = 25\,N\\$

And this results into $25\,N$ which is the correct answer. But can somebody help me here?.

**Why**am I obtaining two different values for the

**force**and for the

**radius**?. My original method did not worked, why?.

Why does it exist a discrepancy between the radius and the maximum speed and the minimum speed?. What am I doing wrong?.