Buoyancy and Object
Buoyancy In physics, buoyancy is a force exerted by a fluid that opposes an object’s weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the column than at the top. This difference in pressure results in a net force that tends to accelerate an object upwards. The magnitude of that force is equal to the difference in the pressure between the top and the bottom of the column, and is also equivalent to the weight of the fluid that would otherwise occupy the column.
For this reason, an object whose density is greater than that of the fluid in which it is submerged tends to sink. If the object is either less dense than the liquid or is shaped appropriately (as in a boat), the force can keep the object afloat. This can occur only in a reference frame which either has a gravitational field or is accelerating due to a force other than gravity defining a “downward” direction (that is, a non-inertial reference frame). In a situation of fluid statics, the net upward buoyancy force is equal to the magnitude of the weight of fluid displaced by the ith the clarifications that for a sunken object the volume of displaced fluid is the volume of the object, and for a floating object on a liquid, the weight of the displaced liquid is the weight of the object. More tersely: Buoyancy = weight of displaced fluid. Archimedes’ principle does not consider the surface tension (capillarity) acting on the body, but this additional force modifies only the amount of fluid displaced, so the principle that Buoyancy = weight of displaced fluid remains valid.
The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). In simple terms, the principle states that the buoyancy force on an object is going to be equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational constant, g. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy. Suppose a rock’s weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting upon it.
Suppose that when the rock is lowered into water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyancy force: 10 ? 3 = 7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor. It is generally easier to lift an object up through the water than it is to pull it out of the water. Assuming Archimedes’ principle to be reformulated as follows, then inserted into the quotient of weights, which has been expanded by the mutual volume yields the formula below.
The density of the immersed object relative to the density of the fluid can easily be calculated without measuring any volumes:(This formula is used for example in describing the measuring principle of a dasymeter and of hydrostatic weighing. ) Example: If you drop wood into water buoyancy will keep it afloat. Example: A helium balloon in a moving car. In increasing speed or driving a curve, the air moves in the opposite direction of the car’s acceleration. The balloon however, is pushed due to buoyancy “out of the way” by the air, and will actually drift in the same direction as the car’s acceleration. Forces and equilibrium
This is the equation to calculate the pressure inside a fluid in equilibrium. The corresponding equilibrium equation is: where f is the force density exerted by some outer field on the fluid, and ? is the stress tensor. In this case the stress tensor is proportional to the identity tensor: Here is the Kronecker delta. Using this the above equation becomes: Assuming the outer force field is conservative, that is it can be written as the negative gradient of some scalar valued function: Then: Therefore, the shape of the open surface of a fluid equals the equipotential plane of the applied outer conservative force field.
Let the z-axis point downward. In this case the field is gravity, so ? = ?? fgz where g is the gravitational acceleration, ? f is the mass density of the fluid. Taking the pressure as zero at the surface, where z is zero, the constant will be zero, so the pressure inside the fluid, when it is subject to gravity, is So pressure increases with depth below the surface of a liquid, as z denotes the distance from the surface of the liquid into it. Any object with a non-zero vertical depth will have different pressures on its top and bottom, with the pressure on the bottom being greater.
This difference in pressure causes the upward buoyancy forces. The buoyancy force exerted on a body can now be calculated easily, since the internal pressure of the fluid is known. The force exerted on the body can be calculated by integrating the stress tensor over the surface of the body which is in contact with the fluid: The surface integral can be transformed into a volume integral with the help of the Gauss divergence theorem: where V is the measure of the volume in contact with the fluid, that is the volume of the submerged part of the body. Since the fluid doesn’t exert force on the part of the body which is outside of it.
The magnitude of buoyancy force may be appreciated a bit more from the following argument. Consider any object of arbitrary shape and volume V surrounded by a liquid. The force the liquid exerts on an object within the liquid is equal to the weight of the liquid with a volume equal to that of the object. This force is applied in a direction opposite to gravitational force, that is of magnitude: where ? f is the density of the fluid, Vdisp is the volume of the displaced body of liquid, and g is the gravitational acceleration at the location in question.
If this volume of liquid is replaced by a solid body of exactly the same shape, the force the liquid exerts on it must be exactly the same as above. In other words the “buoyancy force” on a submerged body is directed in the opposite direction to gravity and is equal in magnitude to The net force on the object must be zero if it is to be a situation of fluid statics such that Archimedes principle is applicable, and is thus the sum of the buoyancy force and the object’s weight If the buoyancy of an (unrestrained and unpowered) object exceeds its weight, it tends to rise.
An object whose weight exceeds its buoyancy tends to sink. Calculation of the upwards force on a submerged object during its accelerating period cannot be done by the Archimedes principle alone; it is necessary to consider dynamics of an object involving buoyancy. Once it fully sinks to the floor of the fluid or rises to the surface and settles, Archimedes principle can be applied alone. For a floating object, only the submerged volume displaces water. For a sunken object, the entire volume displaces water, and there will be an additional force of reaction from the solid floor.
In order for Archimedes’ principle to be used alone, the object in question must be in equilibrium (the sum of the forces on the object must be zero), therefore; and therefore showing that the depth to which a floating object will sink, and the volume of fluid it will displace, is independent of the gravitational field regardless of geographic location. (Note: If the fluid in question is seawater, it will not have the same density (? ) at every location. For this reason, a ship may display a Plimsoll line. ) It can be the case that forces other than just buoyancy and gravity come into play.
This is the case if the object is restrained or if the object sinks to the solid floor. An object which tends to float requires a tension restraint force T in order to remain fully submerged. An object which tends to sink will eventually have a normal force of constraint N exerted upon it by the solid floor. The constraint force can be tension in a spring scale measuring its weight in the fluid, and is how apparent weight is defined. If the object would otherwise float, the tension to restrain it fully submerged is:
When a sinking object settles on the solid floor, it experiences a normal force of: It is common to define a buoyancy mass mb that represents the effective mass of the object as can be measured by a gravitational method. If an object which usually sinks is submerged suspended via a cord from a balance pan, the reference object on the other dry-land pan of the balance will have mass: where is the true (vacuum) mass of the object, and ? o and ? f are the average densities of the object and the surrounding fluid, respectively. Thus, if the two densities are equal, ? o = ? , the object is seemingly weightless, and is said to be neutrally buoyant. If the fluid density is greater than the average density of the object, the object floats; if less, the object sinks. Another possible formula for calculating buoyancy of an object is by finding the apparent weight of that particular object in the air (calculated in Newtons), and apparent weight of that object in the water (in Newtons). To find the force of buoyancy acting on the object when in air, using this particular information, this formula applies: ‘Buoyancy force = weight of object in empty space ? eight of object immersed in fluid’ The final result would be measured in Newtons. Air’s density is very small compared to most solids and liquids. For this reason, the weight of an object in air is approximately the same as its true weight in a vacuum. The buoyancy of air is neglected for most objects during a measurement in air because the error is usually insignificant (typically less than 0. 1% except for objects of very low average density such as a balloon or light foam). Stability A floating object is stable if it tends to restore itself to an equilibrium position after a small displacement.
For example, floating objects will generally have vertical stability, as if the object is pushed down slightly, this will create a greater buoyancy force, which, unbalanced by the weight force, will push the object back up. Rotational stability is of great importance to floating vessels. Given a small angular displacement, the vessel may return to its original position (stable), move away from its original position (unstable), or remain where it is (neutral). Rotational stability depends on the relative lines of action of forces on an object.
The upward buoyancy force on an object acts through the center of buoyancy, being the centroid of the displaced volume of fluid. The weight force on the object acts through its center of gravity. A buoyant object will be stable if the center of gravity is beneath the center of buoyancy because any angular displacement will then produce a ‘righting moment’. Compressible fluids and objects The atmosphere’s density depends upon altitude. As an airship rises in the atmosphere, its buoyancy decreases as the density of the surrounding air decreases.
In contrast, as a submarine expels water from its buoyancy tanks, it rises because its volume is constant (the volume of water it displaces if it is fully submerged) while its mass is decreased. Compressible objects As a floating object rises or falls, the forces external to it change and, as all objects are compressible to some extent or another, so does the object’s volume. Buoyancy depends on volume and so an object’s buoyancy reduces if it is compressed and increases if it expands.
If an object at equilibrium has a compressibility less than that of the surrounding fluid, the object’s equilibrium is stable and it remains at rest. If, however, its compressibility is greater, its equilibrium is then unstable, and it rises and expands on the slightest upward perturbation, or falls and compresses on the slightest downward perturbation. Submarines rise and dive by filling large tanks with seawater. To dive, the tanks are opened to allow air to exhaust out the top of the tanks, while the water flows in from the bottom.
Once the weight has been balanced so the overall density of the submarine is equal to the water around it, it has neutral buoyancy and will remain at that depth. The height of a balloon tends to be stable. As a balloon rises it tends to increase in volume with reducing atmospheric pressure, but the balloon’s cargo does not expand. The average density of the balloon decreases less, therefore, than that of the surrounding air. The balloon’s buoyancy decreases because the weight of the displaced air is reduced. A rising balloon tends to stop rising. Similarly, a sinking balloon tends to stop sinking.