C3.4 · Relate mass, gravitational field strength, and force of gravity

Ontario Grade 11 Physics

Forces

Understanding how an object's mass and the local gravitational field determine its weight

What you will learn

• Distinguish mass, gravitational field strength, and force of gravity.
• Use the relationship between mass, gravitational field strength, and gravitational force.
• Include units, direction, and a reasonableness check in a calculation.

1. Prerequisite bridge: mass, force, and direction

• Mass is a scalar measured in kilograms.
• Gravitational field strength and force of gravity are vectors.
• Near Earth's surface, gravitational field strength is about $9.8\ \mathrm{N/kg}$ downward.

2. The relationship

• Use $F_g=mg$ to calculate the magnitude of gravitational force.
• The force of gravity has the direction of the gravitational field.
• Check that $\mathrm{kg}\times\mathrm{N/kg}=\mathrm{N}$.

3. Choosing a direction and solving

• Define the system and positive direction before solving.
• Report a force as a magnitude and a direction.
• Check units, rounding, and whether the size is reasonable.

4. Rearranging the relationship

• Mass can be found using $m=F_g/g$.
• Field strength can be found using $g=F_g/m$.
• Mass has no direction; gravitational force and field strength do.

Worked example

Finding the force of gravity near Earth's surface

1. Set up
   The system is the toolbox. Choose downward as positive. The known values are its mass and the local field strength; the unknown is the force of gravity.
   $$m=6.2\ \mathrm{kg},\quad g=9.8\ \mathrm{N/kg}$$
2. Apply the relationship
   The magnitude of gravitational force is mass multiplied by field strength. The force points downward because the field points downward.
   $$F_g=mg=(6.2\ \mathrm{kg})(9.8\ \mathrm{N/kg})$$
3. Calculate and check
   The product is $60.76\ \mathrm{N}$. Round to two significant figures, matching the given mass. The units reduce to newtons, and a force of about $61\ \mathrm{N}$ is reasonable for a few kilograms in Earth's field.
   $$F_g\approx61\ \mathrm{N}$$

Worked example

Finding mass from gravitational force

1. Set up
   The system is the object. Choose downward as positive. The force magnitude and field strength are known, and mass is the unknown scalar.
   $$F_g=147\ \mathrm{N},\quad g=9.8\ \mathrm{N/kg}$$
2. Rearrange
   Starting with the force relationship, divide both sides by field strength. The units become kilograms because newtons divided by newtons per kilogram gives kilograms.
   $$m=\frac{F_g}{g}=\frac{147\ \mathrm{N}}{9.8\ \mathrm{N/kg}}$$
3. Calculate and check
   The quotient is $15\ \mathrm{kg}$. The given values each have two significant figures, so the answer is reported to two significant figures. A mass of $15\ \mathrm{kg}$ would experience about $150\ \mathrm{N}$ of force in this field, which is reasonable.
   $$m=15\ \mathrm{kg}$$

Worked example

Finding field strength from force and mass

1. Set up
   The system is the object, and downward is positive. The force magnitude and mass are known. The field strength is the unknown, and its direction is downward because the force points downward.
   $$F_g=12\ \mathrm{N},\quad m=4.0\ \mathrm{kg}$$
2. Rearrange
   Divide the force magnitude by mass to find the force of gravity per kilogram. This gives the field strength's magnitude.
   $$g=\frac{F_g}{m}=\frac{12\ \mathrm{N}}{4.0\ \mathrm{kg}}$$
3. Calculate and check
   The quotient is $3.0\ \mathrm{N/kg}$. It has two significant figures. The value is less than Earth's near-surface value, so it describes a weaker field than the one used in the first example. Its direction remains downward.
   $$g=3.0\ \mathrm{N/kg}$$

Common mistakes and how to avoid them

Lesson summary

• Mass is a scalar measured in kilograms; gravitational field strength is a vector measured in newtons per kilogram.
• The force of gravity is a vector measured in newtons and points in the direction of the gravitational field.
• Use $F_g=mg$ for force, $m=F_g/g$ for mass, and $g=F_g/m$ for field strength.
• Show units and direction, round sensibly, and check that the result makes physical sense.

Check your understanding

Key terms

Scalar

A quantity with size but no direction.

Vector

A quantity with both size and direction.

Mass

A scalar describing how much matter an object contains, measured in kilograms.

Gravitational field

A region where an object with mass experiences a force of gravity.

Gravitational field strength

The force of gravity per kilogram at a location, measured in newtons per kilogram.

Force of gravity

The force exerted on an object with mass by a gravitational field, measured in newtons.

Continue through SPH3U

View the complete SPH3U Ontario Grade 11 Physics curriculum and lessons

• C1.1 · Analyse and improve a technology using Newton’s laws
• C1.2 · Evaluate impacts of technologies that apply forces
• C2.1 · Use force, mass, acceleration, friction, gravity, and normal-force terminology
• C2.2 · Investigate forces with free-body diagrams and Newton’s laws
• C2.3 · Investigate the relationship among net force, mass, and acceleration
• C2.4 · Solve one-dimensional problems with gravity, normal force, and friction

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation C3.4. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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