Coplanar forces equilibrium- a rod in a sphere

In summary, the problem involves a rod inside a smooth sphere, with the rod's center of gravity dividing it into two portions of lengths a and b. The rod is positioned at an angle θ to the horizontal, and subtends an angle 2α at the center of the sphere. The task is to prove that tanθ= (b-a)/(b+a) tanα. To solve this, a force triangle is drawn with the weight of the rod and two normal forces. The normal force on the left side is greater than the one on the right. By drawing the forces onto a diagram and considering the distances a and b, the solution can be found.
  • #1
enadiz
7
0

Homework Statement


A rod whose CG divides it into two portions of length a and b rests inside a smooth sphere in a position inclined to the angle θ to the horizontal. If 2α is the angle it subtends at the centre of the sphere, prove that

tanθ= (b-a)/(b+a) tanα

Picture is attached.

Homework Equations





The Attempt at a Solution



Well I drew the force triangle with the weight of the rod, and the two normal forces. I know the normal force on the left side will be greater than the one on the right... now the thing is I am not quite sure what to do to get tan θ, if you do that with the weight of the rod, what is the adjacent component of that triangle? hmm sorry I am really confused haha..
 

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  • #2
hi enadiz! :smile:

(where have you been for the last five years? :biggrin:)

this is just a geometry question, all you need is a proper diagram :wink:

draw the three forces onto the diagram, and the distances a and b …

what does that give you? :smile:
 

FAQ: Coplanar forces equilibrium- a rod in a sphere

1. What is coplanar forces equilibrium?

Coplanar forces equilibrium refers to a situation in which all the forces acting on an object are in the same plane, and the object is in a state of balance or rest. This means that the net force in any direction is equal to zero.

2. How does a rod in a sphere demonstrate coplanar forces equilibrium?

A rod in a sphere is a common example used to demonstrate coplanar forces equilibrium. In this scenario, the forces acting on the rod, such as the weight of the rod and the tension in the string, are all in the same plane. As a result, the rod remains in a state of balance and does not move.

3. What factors affect coplanar forces equilibrium?

The factors that affect coplanar forces equilibrium include the magnitude, direction, and point of application of the forces, as well as the mass and shape of the object. Any changes to these factors can disrupt the equilibrium and cause the object to move.

4. How can we find the resultant force in a coplanar forces equilibrium system?

The resultant force in a coplanar forces equilibrium system can be found by using vector addition. This involves breaking down each force into its components and then adding them together using vector addition methods, such as the parallelogram or triangle method.

5. What is the importance of coplanar forces equilibrium in real-life applications?

Coplanar forces equilibrium is important in various real-life applications, such as architecture, engineering, and construction. It helps ensure that structures and objects are stable and can withstand external forces without collapsing. It is also used in physics to analyze and understand the behavior of objects under different forces.

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