Why do rubber duckies float? If there was only the force of gravity on the duck, it would fall to the bottom of the bathtub. So there must be another force that keeps it afloat-- this is the buoyant force. In this article, we'll discuss what buoyancy is, why it occurs, and how to calculate it.
Buoyancy - Definition and Meaning
We will define and discuss the meaning of buoyancy.
Buoyancy is the upward force that fluids exert on a fully or partially submerged object.
A fluid is anything that flows, such as a liquid or gas. Because they flow, fluids fill in all the spaces of any container they enter and put pressure on anything they come in contact with. When you fill a bathtub with water, the water presses downwards according to its weight, but it will also press against the sides of the tub (otherwise, if you were to cut a hole in the side of the tub, the water would all just stay in place). If you swim in a pool, the water also exerts pressure on you. When you swim to the bottom of a pool, the farther down you go, the more your ears will feel an increase in pressure. There is more pressure the deeper you go because there is more water above you.
If we draw a cube in water, we could add arrows around it to signify the pressure the water puts on the cube, as in the image above. The pressure on the bottom of the cube is greater than the pressure on the top of the cube. If we were to add all the forces from the pressure together, the horizontal forces would cancel each other out because they're equal in opposite directions. Because the forces of the upward arrows are greater than those of the downward arrows, they would add to result in one single upward force. This upward force is the buoyant force.
The buoyant force is applicable to all fluids, including air, not just water. We simply discuss water the most during this article because it is the easiest to visualize and relate to.
Interatomic Electric Forces cause the Buoyant Force
At an atomic level, fluids are made of atoms that are bonded together. Fluids can move, so when an object is submerged in a fluid, the fluid's atoms and bonds get pushed to the side and will bend around the object, but they still want to be back in their original state. They exert interatomic electric forces to push against the object, which ultimately results in the upward buoyant force.
Archimedes' Principle - The Physical Law of Buoyancy
Below, we will define and discuss Archimedes' Principle.
Archimedes' principle states that the upward buoyant force on a fully or partially submerged object is equal to the weight of the fluid that the object displaced.
This principle was discovered by Archimedes when he took a bath and noticed that the bathwater rose according to how much of his body was in the water. The volume of the submerged part of his body was the same volume as the water that rose out of the way--or in other words, was displaced. In the section above, we determined that fluids exert an upward buoyant force on objects; Archimedes tells us that the magnitude of that force equals the weight of the fluid that the object displaced. We'll discuss why this is, both intuitively and mathematically.
Intuitive Explanation
If we pretend our cube from the example above is made of a weightless plastic filled with water, it will float in equilibrium with the surrounding water because all the water weighs the same. The forces acting on the cube are the downward force of gravity and the upward buoyant force. Because the cube isn't accelerating, due to Newton's Second Law, \( \sum F=ma \), these forces added together equal zero. This means the buoyant force equals the weight of the water in the cube.
Now, what if we switched the cube for a metal cube of the exact same size? The water around the cube wouldn't know it was any different than the water-filled cube, so the buoyant force acting on it would be the same--equal to the weight of water the cube could contain. But now the weight of the cube is greater, so it would fall down to the bottom of the glass. If you picked the cube up off the bottom, it would feel lighter than it really is because of the buoyant force pushing upwards on it.
Mathematical Explanation
Now let's look at how we can explain buoyancy mathematically.
In the image above, we have simplified the forces due to the pressure of the water into a single downward force and a single upward force. Force is equal to pressure, \( P \) times the area the pressure is acting on, \( A \), thus we have
$$F=PA.$$
Next, note that the pressure is equal to the density of the fluid times gravity times the height of the fluid, or
$$P=\rho_\mathrm{f}gh.$$
So, the equation for the force acting on the top of the cube is as follows:
$$F_1=\rho_\mathrm{f}gh_1A,$$
and the force acting on the bottom of the cube is:
$$F_2=\rho_\mathrm{f}gh_2A.$$
To find the buoyant force, we want to find the difference between the force acting on the top and the force acting on the bottom:
$$F_2-F_1=\rho_\mathrm{f}g(h_2-h_1)A.$$
Notice that \( h_2-h_1 \) is just the height of the cube, and by multiplying it by the face of the cube, \( A \), we get the volume of the cube, or rather, the volume of water that the cube displaced. Now we get the following equation for the buoyant force:
$$F_\mathrm{b}=\rho_\mathrm{f}V_\mathrm{f}g.$$
Mass is equal to density times volume,
$$m=\rho{V},$$
so we can substitute the mass of the liquid to replace the density and volume of the liquid:
$$F_\mathrm{b}=m_\mathrm{f}g.$$
Since weight is equal to mass times gravity, this result means that the buoyant force is equal to the weight of the displaced fluid, just as Archimedes said.
Pressure increases as depth in the liquid increases, but that doesn't mean that the buoyant force increases. The height of the object stays the same, so the difference in pressures between the top and bottom of the object stays constant no matter how deep the object is in the fluid. The buoyant force is only dependent on the weight of the liquid displaced and gravity-- not the depth of the object.
Buoyant Force Formula
As was just proven above, Archimedes' principle results in the following formula for buoyancy:
$$F_\mathrm{b}=m_\mathrm{f}g.$$
You can also use the following equation, substituting the mass for density times volume as we described above:
$$F_\mathrm{b}=\rho_\mathrm{f}V_\mathrm{f}g.$$
Both of these equations mean the same thing; which one you use just depends on what information you have. The \( \mathrm{f} \) on the mass, density, and volume variables signifies that you use the mass, density, or volume of the fluid, not of the object.
This is the most important thing to remember about buoyancy and where most mistakes occur.
Let's look at our submerged cube from above. It sinks to the bottom of the water. If each side is \( 0.25\,\mathrm{m} \) long, it has a mass of \( 16\,\mathrm{kg} \), and the density of water is \( 1000\,\mathrm{\frac{kg}{m^3}} \), what is the buoyant force acting on the cube?
Using the second equation for buoyant force, we can plug in the density of the water, the volume of the water displaced by the cube (which in this case is the same as the volume of the cube since we know the cube is fully submerged), and gravity:
\begin{align}F_\mathrm{b} &= \rho_\mathrm{f}V_\mathrm{f}g\\F_\mathrm{b} &= (1000\,\mathrm{\frac{kg}{m^3}})(0.25\,\mathrm{m})^3 (9.81\,\mathrm{\frac{m}{s^2}}) \\F_\mathrm{b} &= 153\,\mathrm{N} \\\end{align}
We can compare this number to the weight of the cube, or gravitational force, to make sure it's fully submerged:
\begin{align}F_\mathrm{g}= & mg \\F_\mathrm{g} = & (16\,\mathrm{kg})(9.81\,\mathrm{\frac{m}{s^2}}) \\F_\mathrm{g} = & 157\,\mathrm{N} \\\end{align}
Since the gravitational force is greater than the buoyant force, the cube is fully submerged, so we know we used the correct volume.
Floating Objects
What if our cube was floating instead of sinking? If we know the object is fully submerged, then we know the volume of the fluid that was displaced by the fluid is the same as the volume of the object. But if it floats, this isn't the case. This is why it's important to remember that the volume you use is that of the fluid displaced by the liquid, and not the volume of the object.
When an object floats in a fluid, the only forces acting on it are the buoyant force and gravitational force. We can see the two forces acting on the floating cube in the image below. Since the cube isn't accelerating, the sum of the two forces equals zero. This means that for objects floating in equilibrium, the buoyant force is equal to the gravitational force (or weight of the object) and buoyancy is neutral.
For objects with an acceleration, such as actively sinking objects, the sum of the forces would equal mass times acceleration rather than zero.
Negative Buoyancy
Besides being neutral, buoyancy can also be negative or positive. To distinguish one from the other, we must look at the temperatures inside vs outside a parcel of water. This provides the density difference needed to determine if a parcel has positive or negative buoyancy. Negative buoyancy is the result of a cooler parcel surrounded by warmer water causing the parcel to sink. Sinking occurs because the force of the parcel's weight is greater than the buoyant force. Positive buoyancy, on the other hand, is the result of a warmer parcel surrounded by cooler water causing the parcel to rise. Rising occurs because the buoyant force is greater than the force of the parcel's weight.
Let's say our same cube from the example above has a mass of \( 13\,\mathrm{kg} \) instead of \( 16\,\mathrm{kg} \). This causes the cube to float, but we don't know how much of it sticks out of the water. What percentage of the cube is below the water?
We can write the same buoyant force equation we used above, but this time we can't use the same volume of the cube, since we don't know how deep the cube is submerged. We will split the volume into the area of the bottom of the cube, \( A \), which we know, multiplied by our unknown height,\( h \):
$$F_\mathrm{b}=\rho(Ah)g$$
We can also set the buoyant force equal to the weight of the object (the mass of the object, \( m_o \), times gravity):
$$F_\mathrm{b}=m_{o}g$$
We'll substitute the second equation into the first so we can solve for our unknown height:
\begin{align}m_{o}g= &\rho(Ah)g \\h= & \frac{m_o}{{\rho}A}\\h= &\frac{13\,\mathrm{kg}}{(1000\,\mathrm{\frac{kg}{m^3}})(0.25\,\mathrm{m})^2}\\\end{align}
Now we have our height of the cube that is submerged:
$$h=0.2\,\mathrm{m}$$
To know how much of the cube is submerged, we can create a ratio between the volume under the water (we'll use a \(\mathrm{w}\) subscript for the variable in the water) and the total volume (we'll use a \(\mathrm{t}\) subscript for the total cube variables):
$$\frac{V_\mathrm{w}}{V_\mathrm{t}}=\frac{Ah_\mathrm{w}}{Ah_\mathrm{t}}$$
The areas cancel out since they are the same, so we can plug in the values for the heights:
$$\frac{V_\mathrm{w}}{V_\mathrm{t}}= \frac{0.2\,\mathrm{m}}{0.25\,\mathrm{m}}= 0.8$$
\( 80\,\%\) of the cube is submerged in the water.
Examples of the Effect of Buoyancy
Some examples of the effect of buoyancy include the following:
- If you hold an air-filled ball underwater and then let go, it will pop up to the surface due to buoyancy.
- You can float easier in saltwater than in freshwater because buoyancy is dependent on the density of the fluid, and saltwater has a higher density.
- Buoyancy causes ships to float.
- Buoyancy causes a helium-filled balloon to rise when let go.
Buoyancy - Key takeaways
- Buoyancy is the upward force that a fluid exerts on a fully or partially submerged object.
- The upward buoyant force on an object is equal to the weight of the fluid that the object displaced , \( F_\mathrm{b}=m_\mathrm{f}g \).
- When finding the buoyant force, always use the mass, or density and volume, of the fluid, rather than of the object.
- When an object is floating in a fluid with no other external forces, the buoyant force is equal to the weight of the object.
- When an object is submerged in fluid, the fluid's atoms and bonds bend out of the way but want to be back in their natural state, so the electric forces between the atoms exert a force against the object. These interatomic forces all added together create one single buoyant force.
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