C. Forces · ForcesMass, Weight and Gravitational Field Strength
C3.4 C2.2 C2.1About 70 minutes
Learning goals- I can distinguish between mass and weight (gravitational force).
- I can use Fg = mg to relate mass, gravitational field strength and gravitational force.
- I can describe how g changes from place to place, on Earth and on other worlds.
- I can explain and calculate apparent weight (the normal force) in an accelerating elevator.
Words to know
mass (m)
The amount of matter in an object and a measure of its inertia. Measured in kilograms; the same everywhere.
weight / gravitational force (Fg)
The force of gravity on an object, in newtons. It depends on where the object is.
gravitational field strength (g)
The gravitational force per kilogram at a location: g = Fg ÷ m. At Earth's surface g ≈ 9.8 N/kg, which also equals 9.8 m/s².
gravitational field
The region around a mass where another mass feels a gravitational force.
apparent weight
The weight you feel, which is the normal force (or the reading on a scale) supporting you.
apparent weightlessness
The feeling of having no weight when the normal force is zero, as in free fall.
Same you, different weight
An astronaut with a mass of 80 kg stands on a scale at the Canadian Space Agency in Longueuil, Québec. Later she steps onto a scale on the Moon. Her mass has not changed: same body, same amount of matter. But the Moon scale shows a much smaller number, because the Moon pulls on each kilogram much more weakly.
In everyday language we say "weight" when we mean mass ("I weigh 60 kg"). In physics the two words mean different things, and today you will keep them straight.
Remember from Grade 6
In Grade 6 you learned that weight is the force of gravity on an object, measured in newtons, and that on Earth weight (N) ≈ mass (kg) × 9.8. Now you will see where the 9.8 comes from and what happens when it changes.
The relationship Fg = mg
The gravitational force on an object is
Fg = m g
- m is the mass in kg.
- g is the gravitational field strength at that place, in N/kg. Near Earth's surface, g = 9.8 N/kg, pointing down toward Earth's centre.
- Fg is the gravitational force (weight) in N.
Notice: Fg = mg looks just like Fnet = ma. If gravity is the only force acting (free fall), Newton's second law gives mg = ma, so a = g. That is why g has two equivalent units: 1 N/kg = 1 m/s². Every object in free fall near Earth accelerates at 9.8 m/s², whatever its mass. (More on that in the lesson Free Fall and Air Resistance.)
| Mass | Weight (gravitational force) |
|---|
| What it is | amount of matter; measure of inertia | force of gravity on the object |
| Type of quantity | scalar | vector (points toward the centre of the planet) |
| SI unit | kilogram (kg) | newton (N) |
| Changes with location? | no | yes: depends on g |
| Measured with | a balance (compares masses) | a spring scale or force sensor |
Why g changes
The value of g depends on the mass of the planet and on your distance from its centre. More planet mass means a stronger pull; more distance means a weaker pull. So:
- On Earth, g is about 9.78 N/kg at the equator and 9.83 N/kg at the poles. Earth spins and bulges a little at the equator, so you are slightly farther from the centre there. It is also slightly smaller on a high mountain.
- At the height of the International Space Station (about 400 km up), g is still about 8.7 N/kg. Astronauts float because they and the station are falling around Earth together, not because gravity is gone.
- The Moon has much less mass than Earth, so its g is only about 1.6 N/kg.
In SPH4U you will calculate g from Newton's law of universal gravitation. In this course, use g = 9.8 N/kg for Earth unless told otherwise.
Worked exampleExample 1: Weight on three worlds
Find the weight of a 60 kg student on Earth (g = 9.8 N/kg), on the Moon (1.6 N/kg) and on Mars (3.7 N/kg). What is the student's mass on Mars?
- Earth: Fg = mg = 60 × 9.8 = 588 N (about 590 N).
- Moon: Fg = 60 × 1.6 = 96 N.
- Mars: Fg = 60 × 3.7 = 222 N (about 220 N).
- Mass does not depend on location: still 60 kg on Mars.
About 590 N, 96 N and 220 N. The mass stays 60 kg.
Worked exampleExample 2: Finding g
A 5.0 kg sample hangs from a spring scale on another planet. The scale reads 18.5 N. Find g there.
- g = Fg ÷ m = 18.5 N ÷ 5.0 kg
- g = 3.7 N/kg, the same as on Mars.
Apparent weight in an elevator
A bathroom scale does not measure gravity directly. It measures how hard it has to push up on you: the normal force FN. When you are not accelerating, FN = Fg and the scale shows your true weight. In an accelerating elevator, Newton's second law gives (up +):
FN − mg = ma, so FN = m(g + a)
- Accelerating up (a positive): FN > mg. You feel heavier.
- Accelerating down (a negative): FN < mg. You feel lighter.
- In free fall (a = −g): FN = 0. You feel weightless, even though gravity still acts. This is apparent weightlessness.
The direction of the acceleration matters, not the direction of motion. An elevator moving down but slowing to stop at the lobby is accelerating up, so you feel heavier.
Worked exampleExample 3: Elevator scale
A 55 kg student stands on a scale in an elevator. Find the scale reading when the elevator (a) moves at constant velocity, (b) accelerates upward at 1.5 m/s², (c) accelerates downward at 1.5 m/s².
- Up is +. FN − mg = ma, so FN = m(g + a).
- (a) a = 0: FN = 55 × 9.8 = 539 N (about 540 N).
- (b) a = +1.5: FN = 55 × (9.8 + 1.5) = 55 × 11.3 = 621.5 N (about 620 N).
- (c) a = −1.5: FN = 55 × (9.8 − 1.5) = 55 × 8.3 = 456.5 N (about 460 N).
- A scale marked in "kg" divides FN by 9.8, so it would show about 55, 63 and 47 "kg".
(a) 539 N, (b) about 620 N, (c) about 460 N.
A 55 kg student on a bathroom scale in an elevator (a = 1.5 m/s² in the middle and right panels). Gravity is the same in every panel. The scale reads the normal force, which changes with the acceleration.Just for funYour bathroom scale is secretly a force meter wearing a kilogram costume. It measures the normal force in newtons, then quietly divides by 9.8 so the number looks like a mass. Take it to the Moon, put a 6 kg sack of potatoes on it, and it would cheerfully announce "1.0 kg".
Hands-on activityWeigh it two ways
You need: a kitchen scale (or bathroom scale), a luggage scale or spring scale if you have one, a bag, and some groceries (e.g. 1 kg bag of sugar, a 2 L milk bag or jug).
- Place each grocery item on the kitchen scale. Record the mass in kg.
- Calculate each item's weight in newtons with Fg = mg.
- If you have a luggage scale, hang the bag of groceries from it while holding it still. Most luggage scales show kg or lb, so convert: weight (N) = reading (kg) × 9.8.
- Now hold the luggage scale and lift the bag quickly upward, then lower it quickly. Watch the reading jump up and drop down. Explain with FT = m(g + a).
- Make a table for one item showing its mass and weight on Earth, the Moon and Mars.
Lift gently so nothing swings into anyone, and keep glass containers out of this activity.
Your turn
1Find the weight of a 2.5 kg cat on Earth.
Fg = 2.5 × 9.8 = 24.5 N (about 25 N).
2A backpack weighs 78.4 N on Earth. Find its mass.
m = Fg ÷ g = 78.4 ÷ 9.8 = 8.0 kg.
3What is the weight of a 1200 kg rover on the Moon?
Fg = 1200 × 1.6 = 1920 N (about 1.9 × 10³ N).
4Explain the difference between mass and weight in one or two sentences.
Mass is the amount of matter (in kg) and is the same everywhere. Weight is the gravitational force on that mass (in N), Fg = mg, and changes when g changes.
5A 70 kg person rides an elevator accelerating up at 2.0 m/s². What does a scale under them read?
FN = m(g + a) = 70 × (9.8 + 2.0) = 826 N (about 830 N).
6Why is g slightly smaller at the equator than at the North Pole?
Earth bulges at the equator, so a person there is farther from Earth's centre, and gravity weakens with distance. (Earth's spin also slightly reduces the effective g at the equator.)
More practice: warm-up
1What is the SI unit of weight?
2What is g at Earth's surface, with units?
3Does your mass change if you go to the Moon?
4Weight of a 10 kg mass on Earth?
5Is weight a scalar or a vector?
A vector (it points toward the centre of the planet).
More practice: on your own
1An astronaut's equipment weighs 160 N on the Moon. Find its mass and its weight on Earth.
m = 160 ÷ 1.6 = 100 kg. On Earth: 100 × 9.8 = 980 N.
2A 40 kg child stands on a scale in an elevator that accelerates down at 2.0 m/s². Find the reading.
FN = 40 × (9.8 − 2.0) = 40 × 7.8 = 312 N (about 310 N).
3An elevator moving down slows to a stop at the ground floor. Do you feel heavier or lighter? Explain.
Heavier. Slowing a downward motion means the acceleration points up, so FN = m(g + a) is larger than mg.
4An object weighs 49 N on Earth and 18.5 N on planet X. Find g on planet X.
m = 49 ÷ 9.8 = 5.0 kg. g = 18.5 ÷ 5.0 = 3.7 N/kg.
5A balance and a spring scale are taken to the Moon. Which one will give the same reading for a 1 kg bag of flour as on Earth? Why?
The balance. It compares the bag with standard masses, and gravity weakens on both sides equally. The spring scale measures force, which drops to about 1.6 N.
Challenge
1A 60 kg student stands on a scale in an elevator. The scale reads 690 N. Find the elevator's acceleration (size and direction). Can you tell which way it is moving?
FN − mg = ma: 690 − 588 = 60a, so a = 102 ÷ 60 = 1.7 m/s² up. We cannot tell the direction of motion: it could be speeding up while moving up, or slowing down while moving down.
2What would a scale read for the student above if the elevator cable broke and the elevator fell freely (ignore friction and air)? Explain why the student would feel weightless even though gravity still acts.
a = −9.8 m/s², so FN = m(g + a) = 60 × 0 = 0 N. The student and the scale fall with the same acceleration, so the scale does not push on the student. Gravity (588 N) still acts, but there is no normal force, which is what we feel as weight. (Real elevators have safety brakes to prevent this.)
3On the ISS, g ≈ 8.7 N/kg. A 75 kg astronaut is on board. What is the gravitational force on her? Why does she float?
Fg = 75 × 8.7 ≈ 650 N (about 89% of her Earth weight). She floats because she and the station are both in free fall around Earth with the same acceleration, so nothing has to push on her: her apparent weight is zero.
Quiz
1. Which quantity is the same on Earth and on the Moon?
2. The weight of a 3.0 kg object on Earth is about:
3. The units N/kg are equivalent to:
4. A 50 kg person is in an elevator accelerating up at 2.0 m/s². The normal force on the person is:
5. Astronauts on the International Space Station float because:
6. Where on Earth's surface is g slightly largest?
Big idea: Mass, gravitational field strength and gravitational force are related by Fg = mg (C3.4). Apparent weight in an elevator applies Newton's second law (C2.2).
How to help: Gently correct everyday language: "That's your mass in kilograms; your weight is about 9.8 times that, in newtons." If you ride an elevator together, ask your teen when they feel heavier or lighter, and why.
Common difficulties:
- Giving weight in kilograms, or mass in newtons.
- Thinking there is no gravity in space.
- Thinking the elevator's direction of motion decides the scale reading. Only the direction of acceleration matters.