Newton's law of gravitation

Examples of Newton's law of gravitation in the following topics:

• Dynamics of UCM

  • Newton's universal law of gravitation states that every particle attracts every other particle with a force along a line joining them.
  • Newton's universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them.
  • For two bodies having masses $m$ and $M$ with a distance $r$ between their centers of mass, the equation for Newton's universal law of gravitation is:
  • Historically, Kepler discovered his 3 laws (called Kepler's law of planetary motion) long before the days of Newton.
  • We shall derive Kepler's third law, starting with Newton's laws of motion and his universal law of gravitation.

• Gravity

  • Gravitational potential energy near the Earth can be expressed with respect to the height from the surface of the Earth.
  • (The surface will be the zero point of the potential energy. ) We can express the potential energy (gravitational potential energy) as:
  • Instead, we must use calculus and the general mathematical definition of work to determine gravitational potential energy.
  • For the computation of the potential energy we can integrate the gravitational force, whose magnitude is given by Newton's law of gravitation (with respect to the distance r between the two bodies).
  • Using that definition, the gravitational potential energy of a system of masses m and M at a distance r using gravitational constant G is:

• Defining Graviational Potential Energy

  • What is the gravitational potential energy of a 1kg block on top of a 1m high table?
  • The strength of a gravitational field varies with location.
  • However, when the change of distance is small in relation to the distances from the center of the source of the gravitational field, this variation in field strength is negligible and we can assume that the force of gravity on a particular object is constant.
  • For the computation of the potential energy, we can integrate the gravitational force, whose magnitude is given by Newton's law of gravitation, with respect to the distance $r$ between the two bodies.
  • Using that definition, the gravitational potential energy of a system of masses $m_1$ and $M_2$ at a distance $r$ using gravitational constant $G$ is

• Weight of the Earth

  • When the bodies have spatial extent, gravitational force is calculated by summing the contributions of point masses which constitute them.
  • Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses, and inversely proportional to the square of the distance between them.
  • In modern language, the law states the following: Every point mass attracts every single other point mass by a force pointing along the line intersecting both points.
  • For points inside a spherically-symmetric distribution of matter, Newton's Shell theorem can be used to find the gravitational force.
  • The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance $r_0$ from the center of the mass distribution:

• The Law of Universal Gravitation

  • While an apple might not have struck Sir Isaac Newton's head as myth suggests, the falling of one did inspire Newton to one of the great discoveries in mechanics: The Law of Universal Gravitation.
  • Theorizing that this force must be proportional to the masses of the two objects involved, and using previous intuition about the inverse-square relationship of the force between the earth and the moon, Newton was able to formulate a general physical law by induction.
  • While Newton was able to articulate his Law of Universal Gravitation and verify it experimentally, he could only calculate the relative gravitational force in comparison to another force.
  • It wasn't until Henry Cavendish's verification of the gravitational constant that the Law of Universal Gravitation received its final algebraic form:
  • Because of the magnitude of $G$, gravitational force is very small unless large masses are involved.

• Momentum, Force, and Newton's Second Law

  • In the most general form, Newton's 2nd law can be written as $F = \frac{dp}{dt}$ .
  • This fact, known as the law of conservation of momentum, is implied by Newton's laws of motion.
  • This statement of Newton's second law of motion includes the more familiar $F_{net} = ma$ as a special case.
  • So for constant mass, Newton's second law of motion becomes
  • Newton's second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass.

• Newton and His Laws

  • Newton's laws of motion describe the relationship between the forces acting on a body and its motion due to those forces.
  • The laws of motion will tell you how quickly the car will move from your pushing.
  • There are three laws of motion:
  • Gravitational Force: a massive body is attracted downward by the gravitational force practiced by the Earth
  • Apply three Newton's laws of motion to relate forces, mass, and acceleration

• Constant Velocity Produces a Straight-Line

  • Recall Newton's first law of motion.
  • The particle could exist in a vacuum far away from any massive bodies (that exert gravitational forces) and electromagnetic fields.
  • If the net force on a particle is zero, then the acceleration is necessarily zero from Newton's second law: F=ma.
  • If is between 0 and 90 degrees, then the component of v parallel to B remains unchanged.

• Motion of the Center of Mass

  • You can see that the Newton's 2nd law applies as if we are describing the motion of a point particle (with mass M) under the influence of the external force.
  • Since the sum of all internal forces will be 0 due to the Newton's 3rd law,
  • For example, when we confine our system to the Earth and the Moon, the gravitational force due to the Sun would be external, while the gravitational force on the Earth due to the Moon (and vice versa) would be internal.
  • Since the gravitational forces between the Earth and the Moon are equal in magnitude and opposite in direction, they will cancel out each other in the sum (see ).
  • Derive the center of mass for the translational motion of a rigid body

• The Third Law: Symmetry in Forces

  • Newton used these laws to explain and explore the motion of physical objects and systems.
  • Newton's three laws are:
  • Newton's third law basically states that for every action, there is an equal and opposite reaction.
  • You have undoubtedly witnessed this law of motion.
  • As your mom if she's clear on Newton's Third.