Problem Statement: A uniform wooden ball of mass m radius r and density is released from rest from the bottom of a vertical cylinder of length which is completely filled with a liquid of density and coefficient of viscosity . The ball reaches the surface of liquid with half of the terminal velocity. Answer the following questions based on the information given above? Explain in details.

Solution:

Terminal Velocity:

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is falling prevents further acceleration.

What is the question asking for?

The problem is asking us to find out some unknown parameters based on the given information. We can follow the steps below to answer the questions:

1. Find out the terminal velocity of the wooden ball in the liquid.
2. Find out the density of the liquid.
3. Find out the coefficient of viscosity of the liquid.

Step 1: Find out the terminal velocity of the wooden ball in the liquid

Given, the wooden ball reaches the surface of liquid with half of the terminal velocity. Therefore, we need to find out the terminal velocity of the wooden ball.

The formula for terminal velocity is:

Vt = (2mg) / (ρACd)

Where,
m = mass of the wooden ball
g = acceleration due to gravity
ρ = density of the liquid
A = surface area of the wooden ball
Cd = coefficient of drag

The surface area of the wooden ball can be calculated as follows:

A = 4πr²

Where,
r = radius of the wooden ball

Now, we can substitute the values in the formula for terminal velocity and solve for Vt.

Vt = (2mg) / (ρACd)
Vt = (2mg) / (ρ(4πr²)Cd)

Step 2: Find out the density of the liquid

We can rearrange the formula for terminal velocity as follows:

ρ = (2mg) / (VtACd)

We know the values of m, g, Vt, A, and Cd. Therefore, we can substitute these values in the above formula and solve for ρ.

ρ = (2mg) / (VtACd)

Step 3: Find out the coefficient of viscosity of the liquid

The Reynolds number (Re) is a dimensionless quantity that is used to predict the nature of fluid flow around objects. The formula for Reynolds number is:

Re = (ρvd) / η

Where,
ρ = density of the liquid
v = velocity of the wooden ball
d = diameter of the wooden ball
η = coefficient of viscosity of the liquid

At half of the terminal velocity, the Reynolds number is 1000. We can substitute the values in the formula for Reynolds number and solve for η.

Re = (ρvd) / η
1000 = (ρVt(2r)) / η

We know the values of ρ, Vt, and r. Therefore, we can solve for η.

η = (ρVt(2r)) / 1000

Final Answer:

1. Terminal velocity of the wooden ball in the liquid is (4mg) / (ρπr²Cd)
2. Density of the liquid is (2