Newtonian mechanics

Gravity and weight

Masses attract each other; near Earth, the weight acting on an object has magnitude P = mg.

A falling apple and the Moon orbiting Earth both experience gravity—the attractive interaction between masses. Near the surface, Earth's gravitational pull on an object is its weight.

F=G\frac{m_1m_2}{r^2},\qquad P=mg

Definition: Gravitational force and weight

Two masses $m_1,m_2$ attract with force $F=Gm_1m_2/r^2$ , where $r$ is center-to-center distance and $G$ is the gravitational constant. Near Earth's surface $g\approx9.8\,m/s^2$ , and weight $P=mg$ points toward Earth's center. Mass is measured in kg; weight is a force in N and depends on location.

Adjust separation and masses; observe inverse-square gravity and a two-body orbit.

Mass is not weight

Mass is an object's property; weight is the gravitational force at a location. On the Moon an object's mass is unchanged but its weight is smaller because $g$ is smaller. At a given location, neglecting air resistance, all freely falling objects have acceleration $g$ , independent of mass.

Example: Weight on the Moon

An instrument has mass 6 kg where $g=1.6\,m/s^2$ . Find its weight.

Solution

$P=mg=6\times1.6=9.6\,N$ . Its mass remains 6 kg.

On Earth, a 3 kg object weighs about 29.4 N with $g=9.8 m/s²$ (or 30 N using rounded $g=10$ ). The gravitational force between two masses becomes one quarter as large when their center separation doubles. Mass, measured in kilograms, does not change when an object moves elsewhere; weight is a force in newtons and changes if $g$ changes.

Example: Weighing a bag of rice

A 4 kg bag is on Earth. Take g=10 m/s². What is its weight?

Solution

P=mg=4×10=40 N; its mass remains 4 kg everywhere.

For the same mass, weight on the Moon is less than on Earth because lunar gravitational acceleration is about 1.6 m/s². A spring scale measures force, so its reading changes with location; a balance compares masses by balancing both sides. Keep mass and weight distinct in calculations.

Quick check

What happens to gravitational force if the center-to-center distance doubles?

Which quantity remains the same when an object is taken from Earth to the Moon?

References

Isaac Newton (1687). Principia Mathematica
Young, Freedman (2019). University Physics