Mechanics problem: A Weight and a Mass Suspended from a String

[smnjech]

TL;DR Summary

I have difficulty solving this problem.

A homogeneous sphere of weight G, radius R is suspended at location 0 together with a weight of weight P in the manner indicated in Fig. 2.5. Determine the angles and the force F acting on the sphere from the hinge of the weight P.

The thing that makes me confused is that in the solution of the problem stated in the book, the angle of α/2 appears and I do not know why.

 

[phinds]

You need to show your work. We can't give help until you show an effort.

 

[smnjech]

You need to show your work. We can't give help until you show an effort.

The main idea that I started with was that there has to be static equilibrium, so the sum of all forces acting on a sphere must give zero. I decomposed the forces into x and y components and for the x component I came up with Tsin(φ)=Fcos(α) and for y component Fsin(α)+G=Tcos(φ). But when I later looked at the solution there was α/2 instead of α and I dont umderstand why.

 

[wrobel]

You need the torque equation with respect to the point O to the whole system and a lot of geometry.

 

[jbriggs444]

The main idea that I started with was that there has to be static equilibrium, so the sum of all forces acting on a sphere must give zero. I decomposed the forces into x and y components and for the x component I came up with Tsin(φ)=Fcos(α) and for y component Fsin(α)+G=Tcos(φ). But when I later looked at the solution there was α/2 instead of α and I dont umderstand why.

I am trying to decode your equations to figure out why you think they apply.

So $F$ is the tension in the right hand cord supporting weight $P$. You seem to think that this force is applied at an angle $\alpha$ below the horizontal to the sphere.

Can you identify the two angles where the cord meets the sphere above its equator and then departs from the sphere at its equator? What is the average of those two angles?

At what angle, on average, does the contact force of cord on sphere act?
With what net force does it act?

 

[Mister T]

The thing that makes me confused is that in the solution of the problem stated in the book, the angle of α/2 appears and I do not know why.

Show us the expression in which $\alpha /2$ appears.

 

[wrobel]

Is $\ell$ a rod that connected to the sphere perpendicular and such that they both form a rigid body?

 

[smnjech]

Is $\ell$ a rod that connected to the sphere perpendicular and such that they both form a rigid body?

Yes it is.

 

[wrobel]

Then it is indeed a simple application of the torque equation.

 

[Lnewqban]

Welcome, @smnjech !

Could you identify the terms T and F in the shown equations?

Consider that the string is always wrapping the sphere, for any position.
The force that the string applies on the sphere is always pointing to its center (perpendicular direction to the tangent line formed by both sides of the string).

What is the angle that each of those sides symmetrically form with that tangent line?

 

[wrobel]

The force that the string applies on the sphere is always pointing to its center

not always but in the absence of friction