DoAssignment.ca

C2.3 · Investigate the relationship among net force, mass, and acceleration

Learn to investigate the relationship among net force, mass, and acceleration through clear examples and targeted practice.

Ontario Grade 11 Physics

Forces

Investigating how a change in force or mass changes an object's acceleration

A push or pull can change an object's motion. The effect depends on the total force acting on the object and on the object's mass. This lesson focuses on investigating that relationship. Before calculating, define the object being studied as the system. Choose a reference frame, meaning the point of view used to describe its motion. For the examples, the system is a cart on a level track, viewed from the stationary track. Right is the positive direction; left is negative. A vector has both size and direction. Force and acceleration are vectors. Mass is a scalar: it has size but no direction. The SI unit of force is the newton (N), of mass is the kilogram (kg), and of acceleration is the metre per second squared (m/s²).

What you will learn

Distinguish scalar quantities from vector quantities relevant to motion.
Use the net-force relationship to connect net force, mass, and acceleration.
Describe a fair investigation of how force and mass affect acceleration.
Calculate acceleration or mass, including units and direction, from given information.

1. Prerequisites: motion and force

A change in velocity is acceleration. Velocity includes direction, so an object can accelerate by speeding up, slowing down, or changing direction. In this lesson, we use acceleration along a straight track. If right is positive, acceleration to the right is positive and acceleration to the left is negative.

A force is a push or pull. More than one force can act on a system at once. The net force is the combined effect of all those forces, taking their directions into account. Forces in opposite directions partly or completely cancel. If the forces balance, the net force is zero.

Mass describes how much matter an object contains. In this relationship, a larger mass needs a larger net force to produce the same acceleration. This does not mean that a large force always produces a large acceleration: the mass also matters.

F_{\mathrm{net}}=\sum F

Force and acceleration are vectors; mass is a scalar.
Use signs to show direction along the chosen axis.
Net force means the combined force on the system.

2. The model and what it predicts

The governing relationship for this lesson is that net force equals mass multiplied by acceleration. The symbols are

F_{\mathrm{net}}

for net force,

m

for mass, and

a

for acceleration. The equation works with a consistent direction convention: the direction of the acceleration matches the direction of the non-zero net force.

Rearrange the equation to find whichever quantity is unknown. To find acceleration, divide net force by mass. To find mass, divide net force by acceleration. Keep units in the calculation. A newton is equivalent to a kilogram metre per second squared, so dividing newtons by kilograms gives acceleration in metres per second squared.

The relationship makes two useful predictions. If mass stays the same and net force increases, acceleration increases in the same proportion. If net force stays the same and mass increases, acceleration decreases. These comparisons are the basis for a fair investigation.

F_{\mathrm{net}}=ma

Use the net force, not just one force, in the relationship.
For a fixed mass, greater net force produces greater acceleration.
For a fixed net force, greater mass produces less acceleration.

3. Investigating the relationship fairly

An investigation uses observations or measurements to test a question. To study the effect of net force, keep the system's mass the same and change the net force. Measure the cart's acceleration for each force. A possible setup uses a cart on a level track, a force sensor, and a motion sensor. This is a proposed procedure, not a report of completed measurements.

To study the effect of mass, keep the net force the same and change the total mass of the moving system. Measure acceleration for each mass. In either investigation, keep other relevant conditions as steady as possible. For example, use the same track and sensor arrangement. Account for forces that oppose the cart's motion when determining net force; a single applied force is not necessarily the net force.

Record the values and units you actually measure. Do not fill a results table with predicted values and describe them as observations. A graph can help reveal a pattern: compare acceleration with net force when mass is held constant, then compare acceleration with mass when net force is held constant. The model predicts increasing acceleration in the first comparison and decreasing acceleration in the second. The graph and measurements provide evidence; the equation helps describe the pattern.

A simulation can help explore the predicted pattern, but its output is simulated rather than physical experimental evidence. State clearly whether results came from physical measurements or from a simulation. In either case, use the chosen setup and recorded values to support your conclusion.

a=\frac{F_{\mathrm{net}}}{m}

Change one chosen factor while keeping the other factor fixed.
Measure acceleration and record units and conditions.
Identify results as measured evidence or simulated output.

4. Applying the model and checking the result

Before substituting values, name the system and state the positive direction. Write the net force with its direction, identify the known values, and identify the unknown. If forces point opposite ways, combine them with signs before using the relationship.

After calculating, check the units and direction. For acceleration, the units should reduce to m/s². Its sign should agree with the direction of net force. Also consider whether the size makes sense: for the same net force, a heavier system should have less acceleration. Report a sensible number of significant figures based on the given values.

m=\frac{F_{\mathrm{net}}}{a}

Combine opposite forces before calculating acceleration.
Include a direction in the final answer for a vector.
Check units, sign, and whether the result matches the predicted pattern.

Worked example

Finding acceleration from a net force

A cart has a mass of 4.0 kg. The net force on it is 12 N to the right. Find its acceleration.

Define the system and direction
The system is the cart, viewed from the stationary track. Choose right as positive. The net force is therefore positive, and acceleration is the unknown.
Use the relationship
The net-force model relates the net force to mass and acceleration. Rearrange it to isolate acceleration.
$a=\frac{F_{\mathrm{net}}}{m}$
Substitute and calculate
Use the positive sign for the rightward force. Dividing newtons by kilograms gives acceleration.
$a=\frac{+12\ \mathrm{N}}{4.0\ \mathrm{kg}}=+3.0\ \mathrm{m/s^2}$

Answer: The cart's acceleration is 3.0 m/s² to the right.

Check: The units reduce from N/kg to m/s². The positive sign means right, matching the net force. The result is reasonable: doubling the mass while keeping this force fixed would reduce the acceleration.

Worked example

Combining forces before finding acceleration

A 3.0 kg cart is pulled with 18 N to the right while a 6.0 N force acts to the left. Find its acceleration.

Define the system and direction
The system is the cart on the stationary track. Right is positive. The rightward force is positive and the leftward force is negative.
Find the net force
Add the forces with their signs. The net force points right because the rightward force is larger.
$F_{\mathrm{net}}=+18\ \mathrm{N}+(-6.0\ \mathrm{N})=+12\ \mathrm{N}$
Calculate acceleration
Use the net force and the cart's mass in the model. Keep the positive sign to show direction.
$a=\frac{+12\ \mathrm{N}}{3.0\ \mathrm{kg}}=+4.0\ \mathrm{m/s^2}$

Answer: The cart accelerates at 4.0 m/s² to the right.

Check: The units are N/kg, equivalent to m/s². The direction is right, the same as the net force. Using only the 18 N pull would overstate the net force because the opposing force must be included.

Worked example

Finding mass from force and acceleration

A system accelerates at 2.5 m/s² to the left when the net force is 20 N to the left. Find its mass.

Define the system and direction
The system is the object being accelerated, viewed from the stationary surroundings. Choose left as positive so both the net force and acceleration are positive. Mass is the unknown.
Rearrange the model
Divide net force by acceleration to isolate mass. Since the directions were assigned consistently, both values have positive signs.
$m=\frac{F_{\mathrm{net}}}{a}$
Substitute and calculate
Substitute the net force and acceleration with their SI units.
$m=\frac{+20\ \mathrm{N}}{+2.5\ \mathrm{m/s^2}}=8.0\ \mathrm{kg}$

Answer: The system's mass is 8.0 kg.

Check: The units N divided by m/s² reduce to kg. Mass is a scalar, so it has no direction. The value is consistent with the relationship: the given force acting on this mass produces the stated acceleration.

Common mistakes and how to avoid them

Using one applied force as though it were always the net force.

Correction: Include all forces on the system and combine their directions before calculating.

Treating mass as a vector or attaching a direction to it.

Correction: Mass is a scalar. Force and acceleration have both magnitude and direction.

Assuming a larger mass always means a larger acceleration.

Correction: Compare situations with the net force specified. At fixed net force, increasing mass reduces acceleration.

Reporting a simulated value as a physical measurement.

Correction: Label simulated output clearly. Only measurements from a physical setup are experimental measurements.

Lesson summary

The system is the object or group of objects being studied; define it and the positive direction before calculating.
Net force combines all forces acting on the system, including their directions.
The relationship is $F_{\mathrm{net}}=ma$ .
At fixed mass, more net force means more acceleration. At fixed net force, more mass means less acceleration.
A fair investigation changes one factor at a time, measures acceleration, and clearly identifies the source of its evidence.

Check your understanding

Question 1

A 2.0 kg object has a net force of 10 N to the right. What is its acceleration?

5.0 m/s² to the right
5.0 m/s² to the left
20 m/s² to the right
0.20 m/s² to the right

Show answer and explanation

5.0 m/s² to the right

Divide net force by mass: 10 N ÷ 2.0 kg = 5.0 m/s². The acceleration points right because the net force points right.

Question 2

The net force stays the same while the mass doubles. What happens to the acceleration?

It doubles.
It is cut in half.
It stays the same.
It becomes zero.

Show answer and explanation

It is cut in half.

With the same net force, acceleration is inversely related to mass. Doubling the mass cuts the acceleration in half.

Question 3

A cart is pulled with 14 N right and 14 N left. What is the net force?

28 N right
14 N left
0 N
28 N left

Show answer and explanation

0 N

The equal forces act in opposite directions and cancel, so their combined net force is zero.

Key terms

Acceleration: The rate at which velocity changes; it has a direction and is measured in m/s².
Mass: A scalar quantity measured in kilograms that describes how much matter is in a system.
Net force: The combined effect of all forces acting on a system, including their directions.
Reference frame: The point of view used to describe an object's position and motion.
System: The object or group of objects selected for study.
Vector: A quantity that has both magnitude and direction.

Continue through SPH3U

View the complete SPH3U Ontario Grade 11 Physics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation C2.3. 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.

Official curriculum reference

Report a correction or ask a question