Find the value of tension just sufficient to achieve limiting equilibrium in the following case:A heavy ring of mass 5kg is threaded on a fixed...

Here we have a ring threaded on a fixed rough horizontal
rod with the coefficient of friction between the rod and the ring being
0.5

A string is tied to the ring and pulled down with a
tension equal to T at 30 degrees to the horizontal. The force on the ring due to the
tension can be divided into two components, one in the vertical direction acting
downwards and another in the horizontal direction.

The
total force on the ring acting downwards is 5*g + T* sin
30.

The force of friction is given by the product of the
coefficient of friction and the normal force = (5*g + T* sin
30)*0.5.

The force acting on the ring trying to move it in
the vertical direction is equal to T*cos 30. This is opposed by the force of friction in
the opposite direction equal to (5*g + T*sin 30)*0.5

When
the two forces are equal the equilibrium is limiting,

T*cos
30 = (5*g + T*sin 30)*0.5

=> T*cos 30 - 0.5*T*sin 30
= 5*g

=> T (cos 30 - 0.5*sin 30) =
5*9.8

=> T = 5*9.8/ (cos 30 - 0.5*sin
30)

=> T = 49/ (sqrt 3 / 2 -
0.25)

=> T =
79.542

Therefore the tension that has to be
applied to reach limiting equilibrium is 79.542
N.