# Equation for a floating solid sphere

• Grand
In summary: Yes, I am familiar with non-linear second order differential equations. In summary, the conversation discusses the equation of motion for a sphere floating in water and the need to include the changing buoyant force as the volume under water changes. The volume of the sphere submerged in water is also calculated as a function of y. The ultimate goal is to find the period of oscillations, which involves solving a non-linear second order differential equation and finding limit cycles.
Grand

## Homework Statement

A sphere is floating in water. It is pushed just under the water level and released. I'm asked to write the equation of motion for the sphere, not assuming small oscillations.

Is it just:
$$my''=\rho V g - mg$$
?

Or do I have to include that the buoyant force is changing while the volume under water changes as well?

Grand said:
Or do I have to include that the buoyant force is changing while the volume under water changes as well?

Yes, you have to include it.

ehild

Any hint on how to do this?

Calculate the volume of the sphere submerged in water as function of y.

ehild

I found that the submerged volume is:
$$V=\frac{4\pi}{3}a^3-\frac{\pi(a-y)^2}{3}(3a-(a-y))=\frac{\pi}{3}(2a^3+3a^2y-y^3)$$
but how do I find the period of oscillations from here on?

The text says that you have to write the equation of motion.
What is y?

ehild

The problem says find the eq of motion and then the period of oscillations (not small) but surely the first step is the equation of motion.

y is the coordinate of the centre of the sphere.

Are you familiar with non-linear second order differential equations? Can you find limit cycles? It is certainly not Introductory Physics.
You can find the period of small oscillation around the equilibrium position of the sphere.

ehild

## What is the equation for a floating solid sphere?

The equation for a floating solid sphere is given by Archimedes' principle, which states that the buoyant force acting on an object is equal to the weight of the fluid it displaces. This can be expressed as Fb = ρfVg, where Fb is the buoyant force, ρf is the density of the fluid, V is the volume of the fluid displaced, and g is the acceleration due to gravity.

## How do you calculate the buoyant force of a floating solid sphere?

To calculate the buoyant force of a floating solid sphere, you need to know the density of the fluid it is floating in, the volume of the fluid displaced by the sphere, and the acceleration due to gravity. The buoyant force can be calculated using the equation Fb = ρfVg, where ρf is the density of the fluid, V is the volume of the fluid displaced, and g is the acceleration due to gravity.

## What factors affect the buoyant force on a floating solid sphere?

The buoyant force on a floating solid sphere is affected by several factors, including the density of the fluid it is floating in, the volume of the fluid displaced by the sphere, and the acceleration due to gravity. Additionally, the shape and size of the sphere can also affect the buoyant force.

## How does the buoyant force on a floating solid sphere compare to its weight?

According to Archimedes' principle, the buoyant force on a floating solid sphere is equal to the weight of the fluid it displaces. This means that if the buoyant force is greater than the weight of the sphere, it will float. If the buoyant force is less than the weight of the sphere, it will sink.

## What is the significance of the equation for a floating solid sphere?

The equation for a floating solid sphere is significant because it helps us understand the relationship between the buoyant force and the weight of an object. It also allows us to calculate the buoyant force and determine whether an object will float or sink when placed in a fluid. This equation has many practical applications, such as in shipbuilding and designing flotation devices.

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