Mechanics question

Question

A heavy ring of mass 5 kg is threaded on a fixed rough horizontal rod. The coefficient of friction between the ring and the rod is 0.5. A light string is attached to the ring and is pulled downwards with a force of magnitude T newtons acting at an angle of 30 degree to the horizontal. Given that the ring is about to slip along the rod, find the value of T. 

Answer 1

Hi Orawan!

 

I assume the 0.5 is the coefficient of static friction, since we are looking at when the ring is just about to slip (which means up to that point, it has not slipped).

 

Since the string is light, we can assume the force of magnitude T that pulls on the end of the string gets transmitted, without diminishment, to the ring.

 

In the horizontal direction, on the ring, at the point where it is just about to slip, there are two force acting:  the horizontal component of tension, and friction.  So we can set up Newton's 2nd law on the ring in the horizontal.  Let's call the direction of the horizontal component of the tension positive:

 

F_net-h = Tx - F_s =  Tcos(30^o) - F_s = ma = 0 (since the ring is not accelerating horizontally)

 

F_s, the force of static friction, is at its maximum if the ring is just about to slip.  So F_s = μ_sN, so 

 

F_net-h = Tcos(30^o) - μ_sN  = 0 --> Tcos(30^o) - (0.5)N = 0

 

This is one equation with two unknowns, so not enough to solve for T yet.  We have to look in the vertical direction, where the forces acting on the ring are gravity (down, negative), the vertical component of tension (down, negative), and the normal force of the rod on the ring (up, positive).  Newton's second law in this direction 

 

F_net-v = N - mg - T_y = 0 (since the ring is not accelerating vertically)

 

So  N - (5 kg)(9.8 m/s²) - Tsin(30^o) = 0 -->  N - 49 - Tsin(30^o) = 0

 

This is a second equation involving the unknowns T and N.  So now we have two equations linking these quantities:

 

Tcos(30^o) - 0.5*N = 0

N - 49 - Tsin(30^o) = 0

 

This constitutes a system of two linear equations with two unknowns, which you can use to solve for T.  I would suggest using the upper equation to write N in terms of T, then substitute that in for N in the bottom equation, making the bottom equation one only in terms of T as an unknown.  Then you can solve for T.

 

If you have some trouble with that, or questions about anything else, or would like to check an answer, just let me know!