Fluid Mechanics and Styrofoam slab

In summary, the first problem involves finding the area of a Styrofoam slab that is floating with a swimmer resting on top in fresh water. The second problem requires finding the rate of water flow through two pipes with different diameters and pressures.
  • #1
vmind
Hi,

I really need help with the following two questions.



1.) A Styrofoam slab has a thickness of 3.8 cm and a density of 303

kg/m^3. When a 73.2 kg swimmer is resting on it, the slab floats

in fresh water with its top at the same level as the water surface.

What is the area of the slab?


2.) A horizontal pipe 18.0 cm in diameter has a smooth reduction to

a pipe 9.00 cm in diameter. If the pressure of the water in the

larger pipe is 9.00e4 Pa and the pressure in the smaller pipe is

6.00e4 Pa, at what rate does water flow through the pipes in kg/s?
 
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  • #2
Give these problems a try, show your work, and you'll get plenty of help.

Hint for #1: equilibrium
Hint for #2: Bernoulli
 
  • #3
Hi,

I have tried the two problems and this is what I have so far.


1.) B(buyoancy)= mass of swimmer *g = p_water * v_displaced water * g

The weight of the slab = 303*(V_slab)*g = buyoancy of water

Then V_slab = (m*g)/(303*g)

V_slab = l*w*h

A_slab = V/h


2.) A_1 (big opening)= pi*(0.09)^2

A_2 (small opening) = pi*(0.0450)^2

A_1* v_1 = A_2 * v_2

v_2 = (A_1/A_2)* v_1

P_1+ 0.5pv_1^2 = P_2 + 0.5pv_2^2

P_1 - P_2 + 0.5p(v_1^2-(A_1/A_2)*v_1)=0


I am not sure how to find the rate of flow of water through

the pipes?
 
  • #4
Problem #1: Think of all the forces that are acting on the "slab + swimmer". The weight of both act downward; this must be balanced by an upward force. The only upward force is the bouyant force which equals the weight of the volume of water displaced.

Problem #2: Solve that last equation for V1. (The second V1 should be V12.) Once you have the speed, calculate the flow rate, which (volume/time) is A1V1. Use the density of water to get the flow rate in terms of mass.
 

1. What is fluid mechanics?

Fluid mechanics is a branch of physics that studies the behavior of fluids, both liquids and gases, and their interactions with forces and boundaries. It involves the study of how fluids move, the forces that act on them, and the effects of these forces on their motion.

2. How does fluid mechanics relate to styrofoam slabs?

In fluid mechanics, styrofoam slabs are often used as a model or demonstration tool to study the behavior of fluids. This is because styrofoam is lightweight and has a low density, making it float on the surface of a fluid. It can also be easily manipulated and used to visualize the effects of forces and fluid flow.

3. What is the role of buoyancy in fluid mechanics and styrofoam slabs?

Buoyancy is a fundamental principle in fluid mechanics that describes the upward force exerted on an object submerged or floating in a fluid. In the case of a styrofoam slab, the buoyant force acting on it is equal to the weight of the fluid it displaces. This is why it floats on the surface of the fluid.

4. Can styrofoam slabs be used to measure fluid flow?

Yes, styrofoam slabs can be used to measure fluid flow. By placing the slab in a fluid flow, its movement and displacement can be used to calculate the velocity and direction of the fluid. This is often used in experiments and simulations to study the behavior of fluids in different scenarios.

5. How does the density of the fluid affect the behavior of a styrofoam slab?

The density of the fluid affects the behavior of a styrofoam slab because it determines the buoyant force acting on the slab. If the density of the fluid is higher than that of the styrofoam, the slab will sink. If the density of the fluid is lower, the slab will float. This is why styrofoam slabs are often used to study the effects of density on fluid behavior.

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