# Sphere falling through viscous material - velcoity calc

• Bakery87
In summary, the conversation discusses the need to determine the velocity of a sphere as it falls through a viscous fluid, taking into account the fluid's density and viscosity, sphere density, and initial velocity. The goal is to find the velocity as a function of time, with the understanding that it will eventually approach the terminal velocity. There is a mention of an exponential term in the equation and the importance of considering the buoyancy of the sphere in the fluid.
Bakery87
I'm dropping a sphere from a known height, and it enters a viscous fluid. I know the initial velocity as it enters the fluid, from there I need the velocity as it falls through the fluid (as a function of time).

I know it should approach it's settling velocity (terminal velocity) and from there I can use stokes law to get the terminal velocity. What I need is the velocity as it approaches that point. Assuming no spin of the sphere. I know the fluid's density and viscosity, sphere density, and initial velocity as it enters the fluid.

I can find this equation for a skydiver falling through the air, but since the air has very little viscosity it does not contribute to the buoyancy of the skydiver.

You don't think about terminal velocity from the start. Just find out the velocity as a function of time. I remember there is an exponential term in it. If you substitute t as infinity in the equation, you get the terminal velocity.

Hint- Its acceleration will not be constant. From Newton's second law Fnet = ma, here a=dv/dt.

## 1. How does the velocity of a falling sphere through a viscous material affect its motion?

The velocity of a falling sphere through a viscous material affects its motion by creating a resistance force, known as drag, that opposes the direction of motion. This drag force increases as the velocity of the sphere increases, ultimately reaching a point where it equals the weight of the sphere and causes it to reach a terminal velocity.

## 2. What factors influence the velocity of a sphere falling through a viscous material?

The velocity of a sphere falling through a viscous material is influenced by the density, size, and shape of the sphere, as well as the viscosity and density of the material it is falling through. Other factors such as the gravitational pull and initial velocity of the sphere also play a role.

## 3. How can the velocity of a falling sphere through a viscous material be calculated?

The velocity of a falling sphere through a viscous material can be calculated using the Stokes' law equation, which takes into account the density and viscosity of the material, the radius and density of the sphere, and the acceleration due to gravity. The equation is v = (2/9)*((p-p_m)gR^2)/η, where v is the velocity, p is the density of the sphere, p_m is the density of the material, g is the acceleration due to gravity, R is the radius of the sphere, and η is the viscosity of the material.

## 4. How does the viscosity of the material affect the velocity of a falling sphere?

The viscosity of the material affects the velocity of a falling sphere by creating a drag force that opposes the motion of the sphere. The higher the viscosity, the greater the drag force and the slower the sphere will fall. Therefore, a higher viscosity material will result in a lower terminal velocity for the sphere.

## 5. What is the significance of calculating the velocity of a sphere falling through a viscous material?

Calculating the velocity of a sphere falling through a viscous material is significant for understanding the behavior of objects in fluids, such as in the fields of fluid dynamics and aerodynamics. It also has practical applications, such as in the design of parachutes and the study of sediment transport in rivers and oceans.

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