Examples of gravitational constant in the following topics:
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- Gravitational energy is the potential energy associated with gravitational force, as work is required to move objects against gravity.
- Near the surface of the Earth, g can be considered constant.
- However, over large variations in distance, the approximation that g is constant is no longer valid.
- Using that definition, the gravitational potential energy of a system of masses m and M at a distance r using gravitational constant G is:
- where K is the constant of integration.
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- 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.
- 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:
- $G$ represents the gravitational constant, which has a value of $6.674\cdot 10^{-11} \text{N}\text{(m/kg)}^2$.
- Because of the magnitude of $G$, gravitational force is very small unless large masses are involved.
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- Since this problem takes place near the surface of the earth, gravitational acceleration is taken to be the constant $g = 9.8 \text{m/}\text{s}^2$, and the gravitational potential energy is $mgh$.
- Note that "height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant.
- However, over large variations in distance, the approximation that $g$ is constant is no longer valid, and we have to use calculus and the general mathematical definition of work to determine gravitational potential energy.
- 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
- where $K$ is the constant of integration.
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- When the bodies have spatial extent, gravitational force is calculated by summing the contributions of point masses which constitute them.
- where $F$ is the force between the masses, $G$ is the gravitational constant, $m_1$ is the first mass, $m_2$ is the second mass and $r$ is the distance between the centers of the masses.
- For points inside a spherically-symmetric distribution of matter, Newton's Shell theorem can be used to find the gravitational force.
- The portion of the mass that is located at radii $r>r_0$ exerts no net gravitational force at the distance $r_0$ from the center.
- Describe how gravitational force is calculated for the bodies with spatial extent
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- An object reaches escape speed when the sum of its kinetic energy and its gravitational potential energy is equal to zero.
- The gravitational potential energy of the spaceship is:
- Where $G$ is the universal gravitational constant ($G = 6.67 \cdot 10^{-11} \text{m}^3\text{kg}^{-1}\text{s}^{-2}$), $M$ is the mass of the planet, $m$ is the mass of the spaceship, and $r$ is the distance of the spaceship from the planet's center of gravity.
- As $r$ goes to infinity, the value of the gravitational potential energy expression goes to 0.
- Calculate the escape speed of an object given its kinetic energy and the gravitational potential energy
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- Conservation of mechanical energy states that the mechanical energy of an isolated system remains constant without friction.
- Conservation of mechanical energy states that the mechanical energy of an isolated system remains constant in time, as long as the system is free of all frictional forces.
- Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy (PE).
- This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces.
- An example of a mechanical system: A satellite is orbiting the Earth only influenced by the conservative gravitational force and the mechanical energy is therefore conserved.
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- At its 2011 meeting, the General Conference on Weights and Measures (CGPM) agreed that the kilogram should be redefined in terms of the Planck constant.
- In scientific terms, 'weight' refers to the gravitational force acting on a given body.
- This measurement changes depending on the gravitational pull of the opposing body.
- For example, a person's weight on the Earth is different than a person's weight on the moon because of the differences in the gravitational pull of each body.
- In contrast, the mass of an object is an intrinsic property and remains the same regardless of gravitational fields.
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- The gravitational force is responsible for artificial satellites orbiting the Earth.
- We shall derive Kepler's third law, starting with Newton's laws of motion and his universal law of gravitation.
- Since $T^2$ is proportional to $r^3$, their ratio is constant.
- (a) An ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci ($f_1$ and $f_2$) is a constant.
- (b) For any closed gravitational orbit, $m$ follows an elliptical path with $M$ at one focus.
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- Tides are the rise and fall of sea levels due to the effects of gravitational forces exerted by the moon and the sun when combined with the rotation of the Earth.
- The tidal force produced by the moon on a small particle located on Earth is the vector difference between the gravitational force exerted by the moon on the particle, and the gravitational force that would be exerted if it were located at the Earth's center of mass.
- Thus, the tidal force depends not on the strength of the lunar gravitational field, but on its gradient (which falls off approximately as the inverse cube of the distance to the originating gravitational body).
- On average, the solar gravitational force on the Earth is 179 times stronger than the lunar, but because the sun is on average 389 times farther from the Earth its field gradient is weaker.
- Schematic of the lunar portion of earth's tides showing (exaggerated) high tides at the sublunar and antipodal points for the hypothetical case of an ocean of constant depth with no land.
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- If an object experiences no net force, then its velocity is constant: the object is either at rest (if its velocity is zero), or it moves in a straight line with constant speed (if its velocity is nonzero).
- The particle could exist in a vacuum far away from any massive bodies (that exert gravitational forces) and electromagnetic fields.