Problem Statement
A float valve regulates the flow of oil (sp. gr. 0.8) into a cistern. The spherical float is 15 cm in diameter. A weightless link AOB carries the float at B and a valve at A. The link is hinged at O, with ∠AOB = 135°. OA = 20 cm and OB = 50 cm. When flow is stopped, AO is vertical and the oil surface is 35 cm below the hinge. A force of 9.81 N is required on the valve to stop the flow. Determine the weight of the float.
Given Data
- Sp. gr. of oil = 0.8, \( \rho_{oil} = 800 \, \text{kg/m}^3 \)
- Float Diameter, D = 15 cm (Radius, R = 7.5 cm = 0.075 m)
- Valve force, \( P = 9.81 \, \text{N} \)
- Length OA = 20 cm = 0.2 m
- Length OB = 50 cm = 0.5 m
- Vertical distance from hinge O to oil surface = 35 cm = 0.35 m
Diagram of the Float Valve
Solution
For the system to be in equilibrium, the moments about the hinge O must balance. The clockwise moment from the valve force P must equal the counter-clockwise moment from the net upward force on the float.
1. Geometry and Float Submergence
When AO is vertical, the angle of OB with the vertical is \(135^\circ - 90^\circ = 45^\circ\). Let \(h\) be the depth of the float's center B below the oil surface.
Since the center of the float is submerged by 0.0035 m (0.35 cm), the total depth of submergence is the radius plus this amount.
2. Buoyant Force on the Float (\(F_B\))
The buoyant force is the weight of the displaced oil. The volume displaced is the volume of the submerged portion of the sphere.
3. Equilibrium of Moments
Let \(W\) be the weight of the float. The net upward force on the float arm is \(F_B - W\). Taking moments about the hinge O:
The weight of the float is approximately \( 1.87 \, \text{N} \).
Explanation of Concepts
Principle of Moments: The float valve system works like a lever. The problem is solved by balancing the moments about the pivot point (the hinge O). A moment is a turning force (Force × Perpendicular Distance). The clockwise moment from the valve force must equal the counter-clockwise moment from the net force on the float.
Net Force on Float: The float is acted upon by two vertical forces: its own weight (\(W\)) acting downwards and the buoyant force (\(F_B\)) from the displaced oil acting upwards. The net force is the difference between these two (\(F_B - W\)).
Buoyant Force on a Sphere: The buoyant force depends on the volume of the submerged part of the sphere. This volume is calculated using the geometric formula for a spherical cap, which depends on the sphere's radius and the depth of submergence.
