Weight (College Board AP® Physics 1: Algebra-Based)

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Ann Howell

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Calculating weight

The gravitational force exerted by an astronomical body on a relatively small nearby object is called weight
- For example, the Earth is the astronomical body that exerts a gravitational force on a person, a relatively small object standing on the surface nearby
An object with mass has the same mass everywhere in the Universe
- Mass is a measure of the amount of matter a body or substance is composed of
- Mass is a measure of an object's inertia
The weight of an object is determined by the magnitude of the gravitational field at the object's position
- A person of mass 70 kg has a greater weight force on Earth than on the Moon because the gravitational field strength on the Earth is greater

Comparing weight on the Earth and Moon

Comparison of weight on Earth and the Moon. A person weighs 700 N on Earth and 114 N on the Moon despite having the same mass of 70 kg in both places. — **A mass in a stronger gravitational field will have a larger weight**

Derived equation

The gravitational force exerted by an astronomical body on a relatively small nearby object is called weight

$Weight = \vec{F_{g}} = m g$

Derivation:

Step 1: Identify the fundamental principle

Newton's second law

${\vec{F}}_{n e t} = m \vec{a}$

Where:
- ${\vec{F}}_{n e t}$ = net force exerted on the system
- $m$ = mass of the system
- $\vec{a}$ = acceleration of the system

Step 2: Apply the specific conditions

When the gravitational force is the only force exerted on an object, the acceleration, $a$ , towards the center of the object is equal to the magnitude of the gravitational field, $g$ of the object
Hence, $a = g$

${\vec{F}}_{n e t} = |F_{g}| = m g$

The gravitational force exerted by an astronomical body on a relatively small nearby object is weight

$Weight = \vec{F_{g}} = m g$

Weightlessness

A system appears weightless when:
- there are no forces exerted on the system
- when the force of gravity is the only force exerted on the system
When there are no forces exerted on the system, then the object must be infinitely far from any other object so a gravitational force is not exerted upon it
- The only place where one object can be infinitely far from another is when both objects are located in space
An infinitely far object could have 'zero weight' when the force of gravity acting on the object is zero
- This is shown by the equations for calculating weight and the gravitational field strength formula

${\vec{F}}_{g} = m g = m (0) = 0$

$g = \frac{G M}{r^{2}} = \frac{G M}{{(\infty)}^{2}} = 0$

Infinitely far objects

Diagram visualizing the universe with stars scattered on a dark background, featuring red and green lines and arrows illustrating concepts related to light and infinity. — **An object in space can be infinitely far from other objects meaning the gravitational force is zero.**

When the force of gravity is the only force exerted on the system, then the system is in free-fall
- An object is in free-fall only when air resistance is negligible

Free-fall

A person is falling with arms outstretched. An arrow points downward indicating weight as the only acting force, with text below: "Negligible air resistance, so the only force acting is weight." — **When there is no air resistance acting on a falling object then it is in free-fall due to the gravitational force.**

On Earth, when an object is in equilibrium, the normal force between the object and the surface is equal to the force of gravity, resulting in a net vertical force of zero
- Weightlessness is experienced when the normal force between an object and a surface is zero, resulting in the object losing physical contact with the surface
Astronauts predominantly experience weightlessness due to being in continuous free fall around the Earth

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