Gravity and its effects on Fluids:fall, Mass, Weight, Thrust and Pressure

Gravity and its Effects on Fluid: We know that a force is required to change the state of motion of an object, meaning that in the absence of an external force, under ideal conditions, a body will continue to remain in its present state forever. The moon revolves around the earth, the earth revolves around the sun, and any object that is thrown vertically upwards always comes down – what is the force responsible for all these phenomena? Newton explained these using the force of gravitation. Let us learn about gravity and its effects on the fluid in detail here.

What is Gravitation?

For objects moving in a circular motion, their direction of motion changes at every point throughout the motion. The change in the direction of motion results in the change of its velocity or acceleration. This force accelerates the objects and keeps them moving in the circular motion directed towards the circle’s centre.

A Centripetal Force acts between the moon and earth as it revolves around the earth. This centripetal force is provided by the gravitational force of attraction between the earth and the moon. In the absence of this force, instead of directing towards the earth at all points in its orbit, the moon would rather move in a uniform path along a straight line.

It is the force of gravitation due to which the apple falls towards the earth, i.e. The earth applies a force on the apple; then, according to newton’s third law, the apple must also apply a force on the earth. But we do not see the earth moving towards the apple.

The same force acts between the sun and all the planets in our solar system, which led Newton to conclude that every object in the universe attracts every other object or all objects in the universe attract each other. This force of attraction between objects is called the Gravitational force.

Universal Law of Gravitation

According to this law, the force of gravitation between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the centre of the two masses. Mathematically, the force of gravitation \({F_g}\) between two masses \(m\) and \(M\) kept at a distance \(r\) can be given as:

\({F_g} = G\frac{{M \times m}}{{{r^2}}}\)

Here, \(G = 6.673 \times {10^{ – 11}}{\rm{N}}{{\rm{m}}^2}{\rm{k}}{{\rm{g}}^{ – 2}}\) is the universal gravitation constant.

The discovery of the Universal Law Of Gravitation helped scientists understand several phenomena which were previously thought to be unrelated like:

i. The force due to which all objects ultimately fall towards the surface of the earth.
ii. The revolution of the moon around the Earth.
iii. The revolution of all the planets in the solar system around the Sun.
iv. The formation of tides due to the Sun and Moon.

Freefall

When an object falls towards the earth, its direction remains the same, but its velocity changes due to the force of gravity applied by the earth. Any change in velocity produces acceleration. The acceleration that is due to the earth’s gravitational force is called acceleration due to gravity. It is represented by the letter \({\rm{g}}\), and its \({\rm{SI}}\) unit is the same as that of acceleration, i.e., \({\rm{m}}/{{\rm{s}}^2}\). When an object falls solely under the effect of gravity, it is said to be infall.

If a body of mass \(m\) fallsly towards the earth from a height \(d\), then by newton’s second law of motion, the force acting on the body, \(F = m \times g\)

This force is the force of gravitation on the mass due to the earth, thus, \(F = G\frac{{M \times m}}{{{d^2}}}\)

Where \(M\) is the mass of the earth.

From above,

\(m \times g = G\frac{{M \times m}}{{{d^2}}}\)

\(g = G\frac{M}{{{d^2}}}\)

For objects kept near or on the earth’s surface, the distance of the object will be equal to the radius of the earth \(R\). Thus, \(g = G\frac{M}{{{R^2}}}\,……(1)\)

Since the earth isn’t a perfect sphere, the value of \(g\) becomes greater at poles than at the equator. This is because the radius of the earth decreases from the equator to the poles. In general, we take the value of \(g\) to be constant.

Substituting the following values in equation \((1)\):

Gravitation constant: \(G = 6.673 \times {10^{ – 11}}{\rm{N}}{{\rm{m}}^2}\;{\rm{k}}{{\rm{g}}^{ – 2}}\)

Mass of earth: \(M = 6. \times {10^{24}}\;{\rm{kg}}\)

The radius of the Earth: \(R = 6.4 \times {10^6}\;{\rm{m}}\)

We get the value of acceleration due to gravity, \(g = 9.8\;{\rm{m}}/{{\rm{s}}^2}\)

The value of acceleration due to gravity is independent of the body’s mass; thus, all objects- hollow, solid, heavy or light will experience the same acceleration.

Since the acceleration due to gravity is almost uniform near the earth’s surface, the equations of motion of the objects fallingly can be obtained by replacing \(a\) by \(-g\) (since acceleration due to gravity always acts towards the earth’s surface).

1. \(v = u – gt\)
2. \(s = ut – 1/2g{t^2}\)
3. \({v^2} – {u^2} = – 2gs\)

Mass

Mass can be defined as the amount of matter in an object. It gives us a quantitative measure of the inertia or the resistance to acceleration under the action of an external force. It is a fundamental property of the matter. It is measured in kilogram\(\left( {{\rm{kg}}} \right)\).

Weight

The weight of an object is the force with which the earth attracts it towards itself. It has both magnitude and direction, and it is directed vertically downwards towards the earth. It is measured in newton\(\left( {{\rm{N}}} \right)\). Mathematically, the weight of an object of mass \(m\) can be given as:

\(W = m \times g\)

Since weight depends on mass and acceleration due to gravity, mass is constant everywhere, but the value of \(g\) varies from place to place. Thus, the weight of an object varies from one place to another, but since we take the value of \(g\) constant near the earth, the weight of an object is assumed to be constant throughout the earth.

Weight on the Moon

The value of acceleration due to gravity will be different on the moon compared to the earth since the radius and mass of the earth is greater than the radius and mass of the moon. From above, we know \(g = G\frac{M}{{{R^2}}}\).

On substituting the values of mass and radius of earth and moon, we get:

\({g_m} = \frac{1}{6}{g_e}\)

Where, \({g_e}\) is the acceleration due to gravity on earth and \({g_m}\) is the acceleration due to gravity on the moon.

Since the acceleration due to gravity on the moon is a sixth of the acceleration due to gravity on earth, the weight of an object is six times less on the moon than its weight on the earth.

Thrust and Pressure

Thrust: Thrust is defined as the force acting on an object perpendicular to its surface. In simple terms, it is the net force acting on an object in a particular direction.

Pressure: The effect of thrust depends upon the area on which it is acting. Pressure is defined as the thrust on a unit area or simply the force per unit area acting on an object in a particular direction.

\({\rm{Pressure}} = \frac{{{\rm{ Thrust }}}}{{{\rm{ Area }}}}\)

The \({\rm{SI}}\) unit of pressure is \({\rm{N/{m}}^2}\) and it is called pascal \({\rm{(Pa)}}\).

Therefore, we can conclude that the pressure exerted will be greater on a smaller area and smaller on a larger area for the same value of force. This is the reason why nails and knives have sharp edges while buildings have wide foundations.

Pressure in Fluids

Solids exert pressure due to their weight, and, similarly, fluids (liquids and gases) have weight, and they exert pressure on the walls of the container in which they are enclosed. The pressure exerted by the fluids is transmitted equally in all directions.

Buoyancy

Buoyancy or upthrust is defined as the upward force exerted by a fluid against the weight of an object that is either partially or fully immersed in it. We know that the force due to gravitational attraction acts on all objects in the downward direction. When an object is kept on the surface of the fluid, its weight pulls it downwards, but the fluid opposes it by applying a force in the upward direction, which is called upthrust or buoyant force.

If the weight of the object kept on the surface of a fluid is greater than the buoyant force, it will sink.

If the weight of the object kept on the surface of a fluid is less than the buoyant force, it will float.

Buoyancy is experienced by all objects immersed in a fluid, and this buoyant force applied by a fluid depends on its density. We know that density is equal to mass per unit volume. Thus, if the object’s density is less than the density of fluid, buoyant force will be large, and it will float on the liquid. But if the object’s density is more than the density of fluid, buoyant force will be small, and it will sink in the liquid.

Archimedes’ Principle

Greek scientist Archimedes studied the buoyant forces acting on different objects as they were immersed in a fluid and stated that “When a given object is completely or partially immersed in a fluid, an upward force acts on it that is equal to the weight of the fluid displaced by it.”

This principle is widely used for designing ships, boats, and submarines. Hydrometers (used for measuring fluid density) and Lactometers( used to determine the purity of milk) are based on the Archimedes Principle.

Relative Density

The ratio of the density of a substance to the density of water is defined as the Relative Density.

\({\rm{Relative}}\,{\rm{density}} = \frac{{{\rm{Density}}\,{\rm{of}}\,{\rm{substance}}}}{{{\rm{Density}}\,{\rm{of}}\,{\rm{water}}}}\)

Relative density is a unitless quantity.

Summary

According to this law, the force of gravitation between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the centre of the two masses. When an object falls solely under the effect of gravity, it is said to be infall. For objects kept near or on the earth’s surface, the distance of the object will be equal to the radius of the earth \(R\).

Thus, \(g = G\frac{M}{{{R^2}}}\) and we get the value of acceleration due to gravity, \(g = 9.8\;{\rm{m}}/{{\rm{s}}^2}\).

The value of acceleration due to gravity is independent of the body’s mass; thus, all objects- hollow, solid, heavy or light will experience the same acceleration.

Fluids (liquids and gases) have weight, and they exert pressure on the walls of the container in which they are enclosed. The pressure exerted by the fluids is transmitted equally in all directions.

When an object is kept on the surface of the fluid, its weight pulls it downwards, but the fluid opposes it by applying a force in the upward direction, which is called upthrust or buoyant force. If the weight of the object kept on the surface of a fluid is greater than the buoyant force, it will sink, and if the weight of the object kept on the surface of a fluid is less than the buoyant force, it will float.

Frequently Asked Questions (FAQs) on Gravity And its Effects on Fluids

Q.1. Define Thrust.
Ans: Thrust is the perpendicular force acting on the surface of an object.

Q.2. Suppose an iron clip and a cork of equal mass are kept on water. Out of these, which one will float on the surface of the water?
Ans: Since the density of cork is less than the density of water, it will float. But the density of iron is greater than the density of water; the clip will sink.

Q.3. Why is the weight of an object lesser on the moon than its weight on the earth?
Ans: The weight of an object is the product of its mass and acceleration due to gravity, and acceleration due to gravity on the moon is one-sixth of the acceleration due to gravity on the earth. Thus, the weight of an object on the moon is one-sixth of its weight on the earth.

Q.4. What is the value of acceleration due to gravity at the centre of the earth?
Ans: The expression for acceleration due to gravity on earth can be given as:
\(g = G\frac{M}{{{R^2}}}\)
Where, \(R\) is the radius of the earth.
At the centre of the earth, \({R = 0}\)
Thus, \(g = G\frac{M}{{{{(0)}^2}}} = 0\)
Acceleration due to gravity at the centre of the earth will be zero.

Q.5.  State universal law of gravitation.
Ans: The universal law of gravitation states that the force of attraction between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Study Newton’s Universal Law of Gravitation

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