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## High school physics

### Course: High school physics>Unit 4

Lesson 4: Newton's law of gravitation

# Newton's law of gravitation review

Review the key concepts, equations, and skills for Newton's law of gravity, including how to find the gravitational field strength.

## Key terms

Term (symbol)Meaning
Gravitational force (F, start subscript, g, end subscript)Attractive force between two objects with mass.
Gravitational fieldA model explaining the influence an object extends to produce a force on other objects.
Gravitational field strength (g)The numerical value of the gravitational field at a point in space. SI units of start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction or start fraction, start text, N, end text, divided by, start text, k, g, end text, end fraction.
Inertial mass (m)Two objects have the same inertial mass if they experience the same acceleration given the same force. This is the same mass used in Newton’s second law. Experimentally equivalent to gravitational mass. Has SI units of start text, k, g, end text.
Gravitational mass (m)The property of matter that causes it to experience a force in a gravitational field. Two objects that balance each other on a scale have the same gravitational mass. Experimentally equivalent to inertial mass. Has SI units of start text, k, g, end text.

## Equations

EquationSymbolsMeaning in words
F, start subscript, g, end subscript, equals, start fraction, G, m, start subscript, 1, end subscript, m, start subscript, 2, end subscript, divided by, r, squared, end fractionF, start subscript, g, end subscript is gravitational force, G is the gravitational constant, m, start subscript, 1, end subscript and m, start subscript, 2, end subscript are the point-like masses, and r is the distance between the massesThe gravitational force between point-like masses m, start subscript, 1, end subscript and m, start subscript, 2, end subscript is directly proportional to their masses and inversely proportional to the square of the distance between them.
g, equals, start fraction, F, start subscript, g, end subscript, divided by, m, start subscript, 2, end subscript, end fraction, equals, start fraction, G, m, start subscript, 1, end subscript, divided by, r, squared, end fractiong is the gravitational field strengthThe gravitational field strength is directly proportional to mass creating the field and inversely proportional to the square of the distance.

## Newton's universal law of gravitation

Gravitational force F, start subscript, g, end subscript is always attractive, and it depends only on the masses involved and the distance between them. Every object in the universe attracts every other object with a force along a line joining them.
The equation for Newton’s law of gravitation is:
F, start subscript, g, end subscript, equals, start fraction, G, m, start subscript, 1, end subscript, m, start subscript, 2, end subscript, divided by, r, squared, end fraction
Where:
F, start subscript, g, end subscript is the gravitational force between m, start subscript, 1, end subscript and m, start subscript, 2, end subscript,
G is the gravitational constant equal to 6, point, 67, times, 10, start superscript, minus, 11, end superscript, start fraction, start text, m, end text, cubed, divided by, start text, k, g, end text, dot, start text, s, end text, squared, end fraction,
and
m, start subscript, 1, end subscript and m, start subscript, 2, end subscript are masses
The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers of mass. This is called an inverse-square law.
For example, if we double the distance between the Earth and the Moon, the attractive force between them would go down (because it is inverse), and it would go down by a factor of 4 instead of 2 (because of the square).
It describes both objects falling down and those in a circular orbit, such as a satellite around Earth.

## How to find the gravitational field strength

All objects attract other objects by producing a gravitational field g, which is defined by the gravitational force per unit mass. We find the strength of this gravitational field of mass m, start subscript, 1, end subscript on any object with mass m, start subscript, 2, end subscript by dividing our above equation by mass m, start subscript, 2, end subscript.
\begin{aligned}g &= \dfrac{F_g}{m_2} \\ \\ &= \dfrac{G m_1 m_2}{r^2 m_2} \\ \\ &= \dfrac{Gm_1}{r^2} \end{aligned}

## Common mistakes and misconceptions

1. Some people forget that gravity causes attraction between all objects. Every mass attracts every other mass. That means you are even gravitationally attracted to your friend, your pet, and even your pizza.
2. People forget that the force of gravity is inversely proportional to r, squared instead of just r. As the distance increases, the force of gravity decreases by a factor of start fraction, 1, divided by, r, squared, end fraction.
3. Sometimes people forget that r is the distance between the centers of mass. We measure the distance between objects from their centers, not their surfaces.

For deeper explanations of Newton's law of gravitation, see our videos:
To check your understanding and work toward mastering these concepts, check out the exercise on gravitational field strength and comparing gravitational and inertial mass.

## Want to join the conversation?

• why is finding the gravity and accelerate Equations so confusing to me ? • Why does the equation for F_g not have a minus sign, since it's in the opposite direction of displacement? • Howdy, so I'm big on proving everything I learn so I was learning how to prove Newton's Law of Gravitation. I started to prove it using Kepler's 3rd Law and everything was going smoothly but then I saw somewhere that you used Newton's Law of Gravitation to prove Kepler's 3rd Law.

While both might be legit, it wouldn't be legit to think in circular reasoning: I must have an alternative way to prove at least one of them.

Newton's 3rd Law: F_g = G*(m1*m2)/(r^2)
Kepler's 3rd Law: P^2 = a^3*(4π^2)/(G(Msun + Mplanet))

So can you provide me with the appropriate background? I'm okay with other proofs that were created after Newton/Kepler but I don't want circular reasoning.

Also I would prefer the original proofs that Newton & Kepler used to prove their laws: unfortunately not near as many people teach proofs as concepts. • Why do we distinguish between inertial mass and gravitational mass? • When an object is thrown straight up, its velocity decreases, and at its maximum height the velocity reduces to zero and then it falls back down. My question is, exactly how long does the object stays in the air at zero velocity, and does it (time) varies from place to place due to different gravity?
(1 vote) • The object will stay at 0 velocity for an infintensimally small time period (it doesn't last long). To visualize this, let's say for example, the object reaches zero velocity 6 seconds when it is thrown. This means that at 5.9999999999999999 seconds, the object still has some velocity. Only at exactly 6 seconds the ball has 0 m/s. At 6.000000000000000000000000001 seconds, the object has velocity (which is really really really close to zero but not exactly zero). This means that only at that small point of time, exactly at 6 seconds (to infinite precision of digits), it will have exactly 0 m/s.

To answer your other question, the time does vary from place to place due to different gravity. For instance, if you would try this same experiment on the Moon, it would take longer for it to fall back since the acceleration due to gravity is slower. (It would also go higher which would mean more distance).

If you are talking about Earth, there are some places where the acceleration due to gravity is different, but that is only due to abnormal events (Earth's buldge, mountains, etc.)
• Hey guys,

Let's say we have a scale (not a balance) made on earth, this scale calculate the weight of an object and then provide it inertial mass, as it give the result in KG, so it must be mass not the weight.

Thus, this scale is probably using an equivalent of the formula of F=mg, while the g is the gravitational field constant on Eatrh.

Therefore, if this scale hasn't a feature to update the g value, it will provide different objects masses on other planets surfaces, and this would be wrong because the inertial mass is constant regardless of the object's location.

If my conclusion is right, the scales manufacturers must inform buyers that their products are only designed to work on Earth, in case someone would want to use them on another planet :) .

If not right, let me know why please!

Thanks
Najm • What is the acceleration experienced by two things (55kg and 17000kg) that are floating out in space? • How do you find the mass of a planet?   