How to obtain the mass of a particle rotating around an axis when the difference of tensions in a wire are known?

Jun 2017
Lima, Peru
The problem is as follows:

A particle of mass $m$ is tied to a very thin wire. Assume the wire is inflexible and of negligible mass. The particle is spinning about a fixed axis as shown located in the center of the circle. Let $T_a$ and $T_b$ be the modulus of the tensions in the string when the particle is located in the points $a$ and $b$ respectively. Find the mass $m$ of the particle if the difference between the tensions $T_{b}-T_{a}=39.2\,N$. Assume $g=9.8\frac{m}{s^2}$

The alternatives are as follows:


For this problem what I've attempted to do was to use the force at the top to be as follows:


At this point the Tension must be zero (I don't know if this statement is correct.)

This reduces the top equation to:


Then to obtain the speed in the lowest point would be by the conservation of mechanical energy:



Then inserting in the above equation would give the speed for the bottom:

Cancelling masses and multiplying by $2$ to both terms:


Since it is known $v_{a}$ then:


Finally I'll use these in the given statements:

The tension in the top:


Tension in the bottom.


Doing a difference between these:


Replacing the known values:





Which results into:


But the answers sheet indicates that the mass is $2\,kg$.

For doing that what it should happenned is that the "$2mg$" is negative in the right side of the equation but I can't find a way to do that. Can someone help me here?. Is it me?, or did I overlooked anything?. Help!. Please.


Math Team
Jul 2011
I agree with your solution ...

$T_b-T_a = \dfrac{m}{r}(v_b^2-v_a^2) + 2mg$

using energy conservation yields $v_b^2-v_a^2= 4rg$