The problem is as follows:
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 will 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?.