Statics - sphere with string attached

[tyche]

I'm really stuck on these two problems:

1) Two equal uniform rods AB BC are smoothly jointed in B and they're in equilibrium. The end of C lies on a rough horizontal plane and the end of A is freely pivoted at a point above the plane. If alpha and beta are the angles of CB and BA with the horizontal, prove that

mu > 2/[tan(beta)+3tan(alpha)]

2)A uniform sphere of radius r and weight W has a ligjht inextensible string attached to a point on its surface, while the other end is attached to a rough vertical wall. The sphere is in equilibrium touching the wall at a distance h below the point of attachment to the wall and is about to slip.
a)If coefficient of friction=mu find the angle of the string with the vertical
b)If mu=h/2r show that the tension in the string is W[Sqrt(1+mu^2)]/2mu


I've tried every combination of forces that came in my mind, with no result.
Of course I've balanced, taken the moments, etc..
Can someone help me? Thanks!

 

[Shooting Star]

In all these kinds of statics problems, the way to proceed is to equate the sum of the horizontal and vertical forces individually to zero; and to equate the moment of all the forces about a conveniently chosen point to zero.

Problem 2:
Let F be the frictional force acting upward at the point of contact, and N the normal reaction acting toward the centre at this point. T is the tension in the string, and makes an angle ‘b’ with the vertical.

Horizontal forces: N=Tsin b .
Vertical forces: W=F+Tcos b.

Taking moment about the point where the string is attached to the wall, Nh=Wr.
Also, if k is the co-eff of friction, then, F=kN.

To find tan b, divide 1st eqn by 2nd, and plug in values of F and N in terms of W, r and h.
To find T, square and add the 1st and 2nd to eliminate b and get T^2.

Problem 1:
For this problem, same approach. The weights of the rods act at their mid-pts. There are two normal reactions at the points A and C, and a horizontal force of friction F at point C. It looks scary but is not. Draw a proper diagram.

 

[ogb p]

I understand that statics deals with the study of objects at rest and the forces acting on them. In the first problem, we are dealing with a system of two rods in equilibrium, with one end of the first rod freely pivoted and the other end of the second rod resting on a rough horizontal surface. The goal is to prove that the coefficient of friction, mu, must be greater than a certain value in order for the system to remain in equilibrium.

To solve this problem, we need to consider the forces acting on the system. First, we have the weight of the rods acting downwards. We also have the normal force from the horizontal surface acting upwards on the end of the second rod. Additionally, we have the tension in the rods due to the pivot and the joint at point B. The angle between the first rod and the horizontal, alpha, and the angle between the second rod and the horizontal, beta, are also important in determining the forces.

Using the conditions for equilibrium, we can set up equations for the sum of forces in the x and y directions. By solving for the tension and plugging it into the equation for the sum of forces in the x-direction, we can simplify the equation to get the desired inequality. This shows that in order for the system to remain in equilibrium, the coefficient of friction must be greater than a certain value.

Moving on to the second problem, we are dealing with a sphere in equilibrium, attached to a vertical wall by a string. The sphere is about to slip, meaning that the forces acting on it are at their maximum limit before it starts to move. We are asked to find the angle of the string with the vertical, as well as the tension in the string, given the coefficient of friction and the dimensions of the sphere.

To solve this problem, we can again use the conditions for equilibrium and set up equations for the sum of forces in the x and y directions. By solving for the tension and plugging it into the equation for the sum of forces in the y-direction, we can get an equation for the angle of the string with the vertical. This shows that the angle is dependent on the coefficient of friction and the dimensions of the sphere.

For part b of the problem, we are given a specific value for the coefficient of friction and asked to find the tension in the string. By substituting this value into the equation for tension, we can simplify and get the desired result. This shows that