A float valve, regulating the flow of water into a reservoir, is shown in the figure. The spherical float (half of the sphere is submerged) is 0.15ab m in diameter. AOB is the weightless link carrying the float at one end, and a valve at the other end which closes the pipe through which flows into the reservoir. The link is mounted on a frictionless hinge at O, and the angle AOB is 135o. The length of OA is 253 mm and the distance between the center of the float and the hinge is 553 mm. When the flow is stopped AO will be vertical. The valve is to be pressed on to the seat with a force of 10,53 N to be completely stop the flow in the reservoir. It was observed that the flow of water is stopped, when the free surface of water in the reservoir is 3ab mm below the hinge at O. Determine the weight of the float sphere.

  9.29 N . . . . weight of 0.948 kg sphere

Explanation:

The sum of torques on the link BOA is zero, so we have ...

  (right force at A)(OA) = (up force at B)(OB·sin(135°))

Solving for the force at B, we have ...

  up force at B = (10.53 N)(253 mm)/((553 mm)/√2) ≈ 6.81301 N

This force is due to the difference between the buoyant force on the float sphere and the weight of the float sphere. Dividing by the acceleration due to gravity, it translates to the difference in mass between the water displaced and the mass of the sphere.

  ∆mass = (6.81301 N)/(9.8 m/s^2) = 0.695205 kg

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The center of the sphere of diameter 0.1553 m is below the waterline by ...

  (553 mm)cos(45°) -(353 mm) = 38.0300 mm

The volume of the spherical cap of radius 155.3/2 = 77.65 mm and height 77.65+38.0300 = 115.680 mm can be found from the formula ...

  V = (π/3)h^3(3r -h) = (π/3)(115.680^2)(3·77.65 -115.68) mm^3 ≈ 1.64336 L

So the mass of water contributing to the buoyant force is 1.64336 kg. For the net upward force to correspond to a mass of 0.695305 kg, the mass of the float sphere must be ...

  1.64336 kg -0.695205 kg ≈ 0.948 kg

The weight of the float sphere is then (9.8 m/s^2)·(0.948 kg) = 9.29 N

The weight of the 0.948 kg float sphere is about 9.29 N.

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explanation: