Gravitation - Gravitational Force and Newton’s Law of Gravitation

What Is Gravitational Force?

Gravitational force is one of the four fundamental forces in nature, along with electromagnetic force, strong nuclear force, and weak nuclear force. It is the force of attraction between any two objects with mass. The greater the mass of an object, the greater its gravitational pull.

Gravitational force is what keeps us on the ground, and it is what holds the planets in orbit around the sun. It is also responsible for the formation of stars and galaxies.

The formula for gravitational force is:

F = Gm1m2/r^2

Where:

• F is the gravitational force in newtons (N)
• G is the gravitational constant (6.674 × 10^-11 N m^2 kg^-2)
• m1 and m2 are the masses of the two objects in kilograms (kg)
• r is the distance between the two objects in meters (m)

For example, the gravitational force between two objects with a mass of 1 kilogram each and a distance of 1 meter between them is:

F = (6.674 × 10^-11 N m^2 kg^-2) * (1 kg) * (1 kg) / (1 m)^2 = 6.674 × 10^-11 N

This is a very small force, but it is enough to keep the two objects from flying apart.

Gravitational force is a very important force in the universe. It is responsible for the structure of the universe and for the way that objects move.

Newton’s Law of Gravitation

Newton’s Law of Gravitation

Sir Isaac Newton’s law of gravitation, published in his Principia Mathematica in 1687, describes the attractive force between any two objects with mass. It is one of the most important and fundamental laws in physics.

The law states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Mathematically, it can be expressed as:

F = Gm1m2/r^2

where:

• F is the force of gravity in newtons (N)
• G is the gravitational constant (6.674 × 10^-11 N m^2 kg^-2)
• m1 and m2 are the masses of the two objects in kilograms (kg)
• r is the distance between the centers of the two objects in meters (m)

Examples:

• The force of gravity between the Earth and the Moon is approximately 2.0 × 10^22 N. This force keeps the Moon in orbit around the Earth.
• The force of gravity between the Sun and the Earth is approximately 3.5 × 10^22 N. This force keeps the Earth in orbit around the Sun.
• The force of gravity between two people standing 1 meter apart is approximately 6.7 × 10^-8 N. This force is too small to be noticeable.

Applications:

Newton’s law of gravitation has many applications in astronomy, engineering, and other fields. Some examples include:

• Calculating the orbits of planets, moons, and other celestial bodies
• Designing spacecraft trajectories
• Determining the mass of planets and stars
• Measuring the Earth’s gravitational field
• Studying the effects of gravity on the human body

Newton’s law of gravitation is a powerful tool that has helped us to understand the universe and our place in it. It is a testament to Newton’s genius and his contributions to science.

Gravitational Force Formula

Gravitational Force Formula

The gravitational force between two objects is given by the formula:

F = Gm1m2/r^2

where:

• F is the gravitational force in newtons (N)
• G is the gravitational constant (6.674 × 10^-11 N m^2 kg^-2)
• m1 and m2 are the masses of the two objects in kilograms (kg)
• r is the distance between the centers of the two objects in meters (m)

Examples

• The gravitational force between the Earth and the Moon is approximately 2.0 × 10^22 N.
• The gravitational force between the Sun and the Earth is approximately 3.5 × 10^22 N.
• The gravitational force between two people standing 1 meter apart is approximately 6.7 × 10^-8 N.

Applications

The gravitational force is responsible for holding the planets in orbit around the Sun, the Moon in orbit around the Earth, and galaxies together. It is also responsible for the tides on Earth.

The gravitational force is one of the four fundamental forces of nature. The other three forces are the electromagnetic force, the strong nuclear force, and the weak nuclear force.

Derivation of Newton’s law of Gravitation from Kepler’s Law

Derivation of Newton’s Law of Gravitation from Kepler’s Laws

Johannes Kepler, a German astronomer, formulated three laws of planetary motion based on his observations of the planets in the solar system. These laws, published in the early 17th century, provided a solid foundation for understanding the dynamics of celestial bodies. Later, Isaac Newton used Kepler’s laws to derive his law of universal gravitation.

Kepler’s Laws

1. Law of Ellipses: Each planet’s orbit around the Sun is an ellipse, with the Sun at one of the two foci of the ellipse.

2. Law of Equal Areas: A line connecting a planet to the Sun sweeps out equal areas in equal time intervals. This means that a planet moves faster when it is closer to the Sun and slower when it is farther away.

3. Law of Harmonies: The square of a planet’s orbital period (the time it takes to complete one orbit) is proportional to the cube of its average distance from the Sun.

Newton’s Law of Gravitation

Newton’s law of gravitation states that every particle of matter in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Mathematically, it can be expressed as:

F = Gm1m2/r^2

where:

• F is the gravitational force between two objects
• G is the gravitational constant (approximately 6.674 × 10^-11 N m^2 kg^-2)
• m1 and m2 are the masses of the two objects
• r is the distance between the centers of the two objects

Derivation of Newton’s Law from Kepler’s Laws

Newton used Kepler’s laws to derive his law of gravitation through a series of mathematical deductions. Here’s a simplified version of the derivation:

1. Consider a planet of mass m orbiting the Sun of mass M in an elliptical path.

2. According to Kepler’s second law, the planet’s areal velocity (the rate at which it sweeps out area) is constant. This means that the planet’s speed is inversely proportional to its distance from the Sun.

3. Let v be the planet’s speed at a distance r from the Sun. Then, the areal velocity can be expressed as:

Areal velocity = (1/2)rv

where A is the area swept out by the line connecting the planet to the Sun in a given time interval.

4. According to Kepler’s third law, the square of the planet’s orbital period (T) is proportional to the cube of its average distance from the Sun (r). Mathematically, it can be written as:

T^2 = Kr^3

where K is a constant.

5. Newton realized that the force acting on the planet to keep it in its orbit must be directed towards the Sun and must be proportional to the planet’s mass (m). He assumed that this force is inversely proportional to the square of the distance between the planet and the Sun (r^2).

6. Equating the gravitational force to the centripetal force required for circular motion, Newton derived the following equation:

F = mv^2/r

where F is the gravitational force.

7. Substituting the expression for areal velocity (1/2)rv into the above equation, Newton obtained:

F = (1/2)m(4π^2r/T^2)

8. Finally, using Kepler’s third law (T^2 = Kr^3), Newton simplified the equation to:

F = Gm1m2/r^2

This equation is identical to Newton’s law of gravitation, where G is the gravitational constant.

Therefore, Newton’s law of gravitation can be derived from Kepler’s laws of planetary motion, demonstrating the connection between empirical observations and theoretical principles in physics.

Solved Examples

Solved Examples

Solved examples are a powerful tool for learning. They provide a concrete illustration of how a concept or principle works, and they can help students to identify and correct their own mistakes.

Here are some examples of solved examples:

• Math: A math teacher might work through a problem on the board, explaining each step as they go. This can help students to understand the process of solving the problem, and it can also help them to identify any areas where they are struggling.
• Science: A science teacher might demonstrate an experiment, and then explain the results. This can help students to understand the concepts that are being taught, and it can also help them to develop their critical thinking skills.
• History: A history teacher might tell a story about a historical event, and then discuss the causes and consequences of the event. This can help students to understand the past, and it can also help them to develop their empathy skills.
• Language arts: A language arts teacher might read a poem or a short story, and then discuss the author’s use of language. This can help students to appreciate literature, and it can also help them to develop their own writing skills.

Solved examples can be a valuable resource for students of all ages. They can help students to learn new concepts, to develop their critical thinking skills, and to improve their academic performance.

Here are some tips for using solved examples effectively:

• Read the example carefully. Make sure that you understand each step of the solution.
• Identify any areas where you are struggling. If you don’t understand something, ask your teacher or a classmate for help.
• Practice solving problems on your own. The more practice you have, the better you will become at solving problems.
• Don’t be afraid to make mistakes. Everyone makes mistakes when they are learning. The important thing is to learn from your mistakes and move on.

Solved examples can be a powerful tool for learning. By using them effectively, you can improve your understanding of the material and your academic performance.

Gravitation Rapid Revision for JEE

Gravitation Rapid Revision for JEE

1. Newton’s Law of Gravitation:

• Every particle of matter in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Mathematically, it can be expressed as:

$$F = Gm_1m_2/r^2$$

Where:

• F is the gravitational force between two objects.
• G is the gravitational constant (6.674 × 10^-11 N m^2 kg^-2).
• m1 and m2 are the masses of the two objects.
• r is the distance between the centers of the two objects.

2. Gravitational Field:

• The gravitational field at a point is defined as the gravitational force experienced by a unit mass placed at that point.

• Mathematically, it can be expressed as:

$$g = F/m$$

Where:

• g is the gravitational field strength.
• F is the gravitational force experienced by the unit mass.
• m is the mass of the unit mass.

3. Gravitational Potential:

• The gravitational potential at a point is defined as the amount of work done in bringing a unit mass from infinity to that point.

• Mathematically, it can be expressed as:

$$V = -Gm/r$$

Where:

• V is the gravitational potential.
• G is the gravitational constant.
• m is the mass of the object creating the gravitational field.
• r is the distance between the point and the center of the object.

4. Kepler’s Laws of Planetary Motion:

• First Law (Law of Ellipses): The orbit of a planet around the Sun is an ellipse, with the Sun at one of the foci.

• Second Law (Law of Equal Areas): A line joining a planet to the Sun sweeps out equal areas in equal intervals of time.

• Third Law (Law of Harmonies): The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

5. Applications of Gravitation:

• Determining the mass of celestial bodies.
• Calculating the force of gravity between objects.
• Predicting the motion of celestial bodies.
• Designing spacecraft trajectories.
• Studying the structure and evolution of the universe.

Examples:

• The gravitational force between the Earth and the Moon is responsible for the Moon’s orbit around the Earth.
• The gravitational force between the Sun and the planets is responsible for the planets’ orbits around the Sun.
• The gravitational force between the Earth and objects on its surface is responsible for the acceleration due to gravity (g = 9.8 m/s^2).
• The gravitational force between galaxies is responsible for the formation of galaxy clusters and superclusters.

Frequently Asked Questions on Gravitation

Will your weight be constant when you are travelling to Greenland from Brazil?

When traveling from Brazil to Greenland, your weight will not remain constant due to variations in gravitational force. Here’s an explanation with an example:

Gravitational Force: Gravitational force is the force of attraction between any two objects with mass. The greater the mass of an object, the stronger its gravitational pull. Earth’s gravitational force keeps us grounded and determines our weight.

Variation in Gravitational Force: Earth’s gravitational force is not uniform throughout the planet. It is stronger at the poles and weaker at the equator. This variation is caused by the Earth’s shape, which is slightly flattened at the poles and bulges at the equator.

Impact on Weight: As you travel from Brazil, which is located near the equator, to Greenland, which is closer to the North Pole, you will experience a change in gravitational force. Greenland has a stronger gravitational pull compared to Brazil.

Example: Consider a person weighing 100 kilograms at sea level in Brazil. When this person travels to Greenland, their weight will increase slightly due to the stronger gravitational force. They might weigh approximately 100.1 kilograms in Greenland.

This difference, though small, demonstrates the impact of varying gravitational force on weight. It’s important to note that while your weight may change slightly during such travel, your mass, which is the amount of matter in your body, remains the same.

In summary, your weight will not be constant when traveling from Brazil to Greenland due to the variation in gravitational force. You will experience a slight increase in weight in Greenland compared to Brazil.

Can you screen the effect of gravitation by any material medium?

Gravitational Shielding

Gravitational shielding refers to the hypothetical possibility of reducing or blocking the effects of gravity using certain materials or methods. According to the theory of general relativity, gravity is not a force but rather a curvature of spacetime caused by the presence of mass and energy. Therefore, it is not possible to completely shield an object from the effects of gravity. However, it is possible to create materials or structures that can reduce the gravitational force acting on an object.

Materials for Gravitational Shielding

There are several materials that have been proposed for use in gravitational shielding. These materials typically have a high density and a low atomic number. Some examples include:

• Lead: Lead is a dense metal that has been used for centuries to shield against radiation. It is also effective at blocking gravitational waves.
• Tungsten: Tungsten is another dense metal that is used in a variety of applications, including radiation shielding and armor. It is also effective at blocking gravitational waves.
• Gold: Gold is a dense metal that is highly resistant to corrosion. It is also effective at blocking gravitational waves.
• Platinum: Platinum is a dense metal that is highly resistant to corrosion. It is also effective at blocking gravitational waves.

Structures for Gravitational Shielding

In addition to materials, there are also several structures that can be used to reduce the effects of gravity. These structures typically involve the use of multiple layers of materials with different densities. Some examples include:

• Spherical shells: A spherical shell is a structure that consists of a hollow sphere made of a dense material. The shell can be used to shield an object from the gravitational forces of other objects outside the shell.
• Cylindrical shells: A cylindrical shell is a structure that consists of a hollow cylinder made of a dense material. The shell can be used to shield an object from the gravitational forces of other objects outside the cylinder.
• Ellipsoidal shells: An ellipsoidal shell is a structure that consists of a hollow ellipsoid made of a dense material. The shell can be used to shield an object from the gravitational forces of other objects outside the ellipsoid.

Applications of Gravitational Shielding

Gravitational shielding has a number of potential applications, including:

• Space travel: Gravitational shielding could be used to protect astronauts from the harmful effects of radiation and gravity during long-duration space missions.
• Medical imaging: Gravitational shielding could be used to improve the quality of medical images by reducing the effects of gravity on the body.
• Military applications: Gravitational shielding could be used to develop new weapons and defense systems.

Challenges of Gravitational Shielding

There are a number of challenges associated with gravitational shielding, including:

• The high density of materials required: The materials used for gravitational shielding must be very dense, which can make them difficult to work with and expensive to produce.
• The need for multiple layers: In order to be effective, gravitational shielding typically requires multiple layers of materials, which can add to the weight and complexity of the structure.
• The difficulty of shielding against all gravitational forces: It is not possible to completely shield an object from the effects of gravity. However, it is possible to reduce the gravitational force acting on an object by using a combination of materials and structures.

Conclusion

Gravitational shielding is a promising technology with a number of potential applications. However, there are a number of challenges associated with gravitational shielding that need to be overcome before it can be widely used.

Why are space rockets launched eastward?

Space rockets are primarily launched eastward due to several factors related to Earth’s rotation and orbital mechanics. Here’s a more in-depth explanation:

Earth’s Rotation:

1. Coriolis Effect: Earth’s rotation creates a phenomenon called the Coriolis effect, which deflects moving objects in a curved path. When a rocket is launched eastward, it benefits from this deflection, as it gains additional velocity from Earth’s rotation. This extra velocity helps the rocket achieve orbit more efficiently.

2. Conservation of Angular Momentum: As a rocket moves eastward, it inherits some of Earth’s angular momentum. This means that the rocket starts its ascent with a higher initial velocity, reducing the amount of energy required to reach orbit.

Orbital Mechanics:

1. Gravitational Pull: Earth’s gravitational pull is strongest at the equator and weakest at the poles. By launching eastward from near the equator, rockets take advantage of the reduced gravitational pull, making it easier to overcome Earth’s gravity and enter orbit.

2. Orbital Plane: Launching eastward allows rockets to enter an orbit that is aligned with Earth’s natural rotation. This is important for various reasons, including satellite communication, weather forecasting, and space exploration missions.

Examples:

1. Kennedy Space Center: The Kennedy Space Center in Florida, USA, is located at 28.5 degrees north latitude. Rockets launched from this site benefit from Earth’s rotation and the Coriolis effect, making it an ideal location for eastward launches.

2. Baikonur Cosmodrome: The Baikonur Cosmodrome in Kazakhstan is located at 45.6 degrees north latitude. While it is not as close to the equator as Kennedy Space Center, Baikonur still benefits from eastward launches due to the Coriolis effect and the reduced gravitational pull at higher latitudes.

In summary, space rockets are launched eastward to take advantage of Earth’s rotation and orbital mechanics. This allows rockets to gain additional velocity, reduce the energy required to reach orbit, and enter an orbit aligned with Earth’s natural rotation.

Why does a bouncing ball bounce higher on hills than on planes?

When a ball bounces on a flat surface, it loses some of its energy due to friction and deformation. However, when a ball bounces on a hill, it can gain energy due to the force of gravity. This is because the ball’s potential energy is converted into kinetic energy as it rolls down the hill.

The amount of energy that a ball gains when it bounces on a hill depends on the steepness of the hill. The steeper the hill, the more potential energy the ball will have, and the higher it will bounce.

In addition to the steepness of the hill, the coefficient of restitution of the ball also affects how high it will bounce. The coefficient of restitution is a measure of how elastic a ball is. A ball with a high coefficient of restitution will bounce higher than a ball with a low coefficient of restitution.

Here are some examples of how the steepness of the hill and the coefficient of restitution affect the height of a bouncing ball:

• A ball with a high coefficient of restitution will bounce higher on a steep hill than a ball with a low coefficient of restitution.
• A ball with a low coefficient of restitution will bounce higher on a flat surface than a ball with a high coefficient of restitution.
• A ball will bounce higher on a steep hill than on a flat surface, regardless of its coefficient of restitution.

The following table shows the height of a bouncing ball for different combinations of the steepness of the hill and the coefficient of restitution:

As you can see from the table, the height of a bouncing ball increases with both the steepness of the hill and the coefficient of restitution.

The gravitational potential energy is negative. Why?

The gravitational potential energy of an object is negative because it represents the amount of work that would be required to move the object from its current position to a reference point, typically taken to be infinity. This work is done against the gravitational force, which is always attractive and acts to pull objects towards each other.

To understand why the gravitational potential energy is negative, consider the following example. Imagine a ball of mass m held at a height h above the ground. The gravitational force acting on the ball is given by:

$$F_g = mg$$

where g is the acceleration due to gravity.

The work done to lift the ball from the ground to a height h is given by:

$$W = F_g * h = mgh$$

This work is stored as gravitational potential energy in the ball. The gravitational potential energy of the ball is given by:

$$U_g = mgh$$

The gravitational potential energy is negative because the work done to lift the ball is negative. This is because the gravitational force is acting in the opposite direction to the displacement of the ball.

The negative sign of the gravitational potential energy indicates that the ball is in a bound state. This means that the ball will fall back to the ground if it is released. The amount of work that is required to lift the ball to a given height is equal to the amount of work that the ball will do when it falls back to the ground.

The gravitational potential energy is a fundamental concept in physics. It is used to describe the motion of objects in gravitational fields. The negative sign of the gravitational potential energy is a consequence of the attractive nature of the gravitational force.

Why is Newton’s law of gravitation called universal law?

Newton’s law of gravitation is called a universal law because it applies to all objects in the universe, regardless of their mass or composition. This means that the same law that governs the motion of planets around the sun also governs the motion of apples falling from trees and the tides of the ocean.

The universality of Newton’s law of gravitation is one of its most important features. It means that scientists can use the same law to explain a wide variety of phenomena, from the motion of planets to the behavior of galaxies. This makes Newton’s law of gravitation a powerful tool for understanding the universe.

Here are some examples of how Newton’s law of gravitation is applied to different objects in the universe:

• Planets: Newton’s law of gravitation explains why planets orbit the sun. The sun’s gravity pulls on the planets, causing them to move in a circular path around it. The closer a planet is to the sun, the stronger the gravitational pull and the faster the planet orbits.
• Moons: Newton’s law of gravitation also explains why moons orbit planets. The gravity of a planet pulls on its moons, causing them to move in a circular path around it. The closer a moon is to its planet, the stronger the gravitational pull and the faster the moon orbits.
• Stars: Newton’s law of gravitation explains why stars form and why they stay together. The gravity of a star pulls on its own matter, causing it to collapse inward. This collapse creates heat and pressure, which eventually ignite nuclear fusion. The outward pressure of nuclear fusion counteracts the inward pull of gravity, causing the star to reach a state of equilibrium.
• Galaxies: Newton’s law of gravitation explains why galaxies form and why they stay together. The gravity of a galaxy pulls on its own stars, causing them to move in a circular path around its center. The closer a star is to the center of the galaxy, the stronger the gravitational pull and the faster the star orbits.

Newton’s law of gravitation is a fundamental law of nature that has been used to explain a wide variety of phenomena in the universe. It is a powerful tool for understanding the universe and how it works.

What is the weight of the body at the centre of the Earth?

At the centre of the Earth, the weight of a body becomes zero. This is because the gravitational force acting on the body from all sides is equal and opposite, resulting in a net force of zero.

To understand this concept, consider the following:

1. Gravitational Force: The gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. At the centre of the Earth, the distance between the body and all the surrounding mass is the same.

2. Equal and Opposite Forces: Due to the spherical symmetry of the Earth, the gravitational forces acting on the body from all sides are equal in magnitude but opposite in direction. This means that the net gravitational force acting on the body is zero.

3. Weight as a Force: Weight is the force exerted on an object due to gravity. Since the net gravitational force at the centre of the Earth is zero, the weight of the body also becomes zero.

4. Example: Imagine a person standing at the centre of the Earth. The gravitational force pulling them towards one side of the Earth is exactly balanced by the gravitational force pulling them towards the opposite side. As a result, the person would experience weightlessness.

It’s important to note that this concept applies specifically to the centre of the Earth. As you move away from the centre, the gravitational force and weight gradually increase due to the decrease in distance from the surrounding mass.

Does friction arise due to gravitation?

Does Friction Arise Due to Gravitation?

Friction is the force that opposes the relative motion of two objects in contact. It is caused by the interaction of the microscopic irregularities on the surfaces of the two objects. When these irregularities come into contact, they create a resistance to motion.

Gravitation is the force of attraction between two objects with mass. It is a fundamental force of nature, and it is what keeps the planets in orbit around the sun.

So, does friction arise due to gravitation? The answer is no. Friction is caused by the interaction of the microscopic irregularities on the surfaces of two objects, while gravitation is the force of attraction between two objects with mass.

Examples

Here are some examples of friction and gravitation:

• When you rub your hands together, you feel the force of friction. This is because the microscopic irregularities on the surfaces of your hands are interacting with each other.
• When you drop a ball, it falls to the ground due to the force of gravitation. This is because the Earth’s gravity is pulling the ball towards it.

Conclusion

Friction and gravitation are two different forces. Friction is caused by the interaction of the microscopic irregularities on the surfaces of two objects, while gravitation is the force of attraction between two objects with mass.