Inside a fixed hollow cyoinder of radius R whose axis is horizontal are placed symmetrically two equal cylinder each of radius r. If third identical is placed symetrically on them, then find maximum value of R for which the three cylinder are in equilibirium.?

Question

Answer

**Problem Statement:**
Inside a fixed hollow cylinder of radius R whose axis is horizontal are placed symmetrically two equal cylinders each of radius r. If a third identical cylinder is placed symmetrically on them, then find the maximum value of R for which the three cylinders are in equilibrium.

**Solution:**
Let's assume the mass of each cylinder is m and their height is h.

**Step 1: Free Body Diagram**
The first step is to draw the free body diagram of the cylinders. The forces acting on the cylinders are weight and normal force.

**Step 2: Equations of Equilibrium**
To find the maximum value of R for which the three cylinders are in equilibrium, we need to write the equations of equilibrium for each cylinder.

For the top cylinder, the equations of equilibrium are:

- ΣFx = 0 (Horizontal equilibrium)
- ΣFy = 0 (Vertical equilibrium)
- ΣM = 0 (Rotational equilibrium)

For the two bottom cylinders, the equations of equilibrium are:

- ΣFx = 0 (Horizontal equilibrium)
- ΣFy = 0 (Vertical equilibrium)

**Step 3: Solving Equations of Equilibrium**
Solving the equations of equilibrium, we get:

- ΣFx = 0: T1 + T2 = 0
- ΣFy = 0: 3mg - N1 - N2 = 0
- ΣM = 0: T1h - T2h = 0

where T1 and T2 are the tensions in the strings holding the top cylinder in place, N1 and N2 are the normal forces acting on the bottom cylinders.

Solving these equations, we get:

- T1 = T2 = mg
- N1 = N2 = 3mg/2

**Step 4: Finding the Maximum Value of R**
To find the maximum value of R, we need to balance the torque acting on the top cylinder. The torque is given by:

- ΣM = (N1 - N2)R = (3mg/2 - 3mg/2)R = 0

Hence, the maximum value of R for which the three cylinders are in equilibrium is infinity. This means that the cylinders can be in equilibrium for any value of R.

Answer

R(1+2√7)

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