How Much Work is Needed to Pull a Half-Submerged Ball Out of Water?

[Grotessque]

Homework Statement

A uniform ball of radius r = 0,3m swim in the water so the half of the ball is in the water. What is the work W required to pull out the ball from the water. (You can assume that the density of the water is ρ = 1000 kg/m³, the acceleration of gravity g = 9,81 m/s², ignore air and fluid resistance).

Homework Equations

- According to hydrostatic law and equilibrium's law the weight force G = mg is equal to lift force F = Vρg, V is volume of the ball inside the water, V =2/3r³π.

G =2/3r³πρg .

- Work W is W = ʃ[mg - ρg(πh²R - 1/3πh³)]dh, (h =0 to h = r)
Expression πh²R - 1/3πh³ represent the volume of ball immersed in water during the process of pulling out the ball.

The Attempt at a Solution

- W =7/12r³πρgr.
W = 145,54J.

 

[anathema]

This solution seems correct. To verify, we can plug in the given values:

- r = 0.3m
- ρ = 1000 kg/m³
- g = 9.81 m/s²

Plugging these values into the equation, we get:

W = (7/12)(0.3³)(π)(1000)(9.81)(0.3)
W = 145.54 J

Therefore, the work required to pull out the ball from the water is approximately 145.54 Joules.

 

[Moetasim]

I would like to clarify that the solution provided is an approximation and may not be entirely accurate due to the assumptions made. While the hydrostatic law and equilibrium's law provide a good estimate of the work required to pull out the ball, there are other factors that could affect the actual work required. These could include the viscosity of the water, the shape of the ball, and the speed at which the ball is pulled out. Additionally, the formula used assumes that the ball is a perfect sphere and that the water is incompressible, which may not always be the case. Furthermore, the work calculated is only for the process of pulling out the ball from the water and does not take into account the work required to initially submerge the ball in the water. Therefore, while the provided solution may provide a good estimate, it is important to keep in mind that it may not be entirely accurate in real-world situations. Further research and experimentation may be needed to obtain a more precise answer.