Here is the problem:
"A ring of mass $3\,\text{kg}$ is at rest on a rough horizontal wire. It is attached to a string that is at an angle of $60^\circ$ above the horizontal. The coefficient of friction between the ring and the wire is $0.7.$ Find the set of values for the tension, $T,$ which will allow the ring to remain in equilibrium."
So I tried using the equation that frictional force equals the horizontal component of the tension of the string acting on the ring. (Used the condition for limiting equilibrium.)
I found the expression for normal force to be, $R=30-T\sin(60^\circ).$ And formed the equation, $T\cos(60^\circ)= R(0.7).$ The tension from this equation is about $198\,\text{N}.$
However this is only one part of the solution. They asked for a set of values. The real answer is : $34.6\le T\le 198.$
Help would be appreciated, thank you.
Best Answer
I think the idea is that the wire is threaded through the ring so that it is the bottom of the wire that makes contact with the inside of the ring. This makes the normal force on the ring point downwards so $R=30-T\sin(60°)<0$ so $$T>30\csc(60°)=34.64\text{ N}$$ Clearly this is a math problem, not a physics problem because $g=10\,m/s^2$.