How to Calculate Force Exerted on Rain Gauge Lid per ml of Rain Water?

[rohit405]

Homework Statement

The .png file attached shows a simple rain gauge where rain water is collected through a funnel of 25cm diameter. As water flows through the sides of the vessel and it fills the vessel it exerts an upward force on the bottom surface of the float. This force is transmitted to the top lid of the vessel(not shown in fig.). Given diameter of the collecting vessel is 13cm and that of the float is 12cm, calculate the force that will be exerted on the lid of the vessel per ml of rain water collected. Neglect weight of the float. Given: volume of float= 1 litre.

Homework Equations

The Attempt at a Solution

I have calculated the height of the float and found it to be 8.84cm.
(Since volume of float is 1litre= 0.001m^3 and dia of float is 12cm, we can find volume)
After this I am confused as to what to do since the water which will pour into the vessel will fill up and push the float upward till it comes in contact with the lid. This means a force will be exerted on the lid and which is to be calculated. How will this vary with increase per ml of rain water filled? Please help!

 

[RTW69]

Is the float filled with air? Can we neglect the weight of the float? Is the float made of a light weight material like Styrofoam? This is a buoyant force problem.

 

[rohit405]

Yes, the weight of the float can be neglected.

 

[RTW69]

It turns out that none of the questions I asked above is relevant to the problem. Essentially you have a float that wants to ride on top of the water but can't because there is a lid keeping it against the bottom of the collection vessel. The force pushing down on the lid will equal to the Buoyant force trying to push the force up. The rain water can only fill the volume between the float and collection vessel. You will need to find out how to calculate the buoyant force of a floating object. The challenge is determining the depth of water in the space between the float and collection vessel for every 1 ml of rain water. The force will indeed change as the vessel fills up but F/ml will be a constant (actually the slope of a line).

 

[DeeNos]

To calculate the force exerted on the lid of the rain gauge per ml of rain water collected, we can use the equation F=PA, where F is the force, P is the pressure, and A is the area over which the force is applied. In this case, the area is the surface area of the float in contact with the lid, which is the same as the area of the lid itself.

To determine the pressure, we can use the equation P = ρgh, where ρ is the density of water, g is the acceleration due to gravity, and h is the height of the float. As you have correctly calculated, the height of the float is 8.84cm.

Substituting these values, we get P = (1000 kg/m^3)(9.8 m/s^2)(0.0884 m) = 863.12 Pa.

Now, to find the force, we need to multiply the pressure by the area, which is the surface area of the lid. This can be calculated using the formula A = πr^2, where r is the radius of the lid. Since the diameter of the collecting vessel is 13cm, the radius is 6.5cm.

Therefore, the force exerted on the lid per ml of rain water collected is F = 863.12 Pa x π(6.5cm)^2 = 176.7 N.

As the volume of rain water collected increases, the force exerted on the lid will also increase proportionally. This is because the height of the float will increase, resulting in a higher pressure and therefore a higher force on the lid.

I hope this helps to clarify your confusion. Let me know if you have any further questions.