B2.4 Predict and investigate forces acting on systems of objects

Ontario Grade 12 Physics

Dynamics

Ontario Grade 12 Physics · Study topic B2.4

In earlier physics, you used motion to describe how objects move and force diagrams to represent pushes and pulls. A force is a vector: it has magnitude and direction. Mass is a scalar: it has magnitude but no direction. A system is the object or group of objects selected for analysis. A reference frame is the viewpoint used to describe motion; here, use the room or laboratory as at rest. State a positive direction before using signs. This lesson focuses on predicting and investigating forces acting on systems of objects.

What you will learn

Choose a system and distinguish external forces from internal interactions.
Draw force diagrams using a stated reference frame and positive direction.
Use the net-force relationship to predict acceleration or an unknown force.
Plan an investigation that compares a prediction with measured evidence.

1. Choose the system and map the forces

Start by naming the object or objects in the system. A force from an object outside the chosen system is an external force. A force between objects both included in the system is an internal interaction. The distinction depends on your system choice. For example, contact between two blocks is internal if both blocks are the system, but external if you analyse one block alone.

A free-body diagram shows external forces acting on one chosen object or system. Represent the system with a simple shape. Draw and label an arrow for each force. The arrow points in the force’s direction; a scaled drawing can also show relative magnitude. Do not draw forces exerted by the system on other objects. Draw forces acting on the system.

Common forces include weight, the gravitational force from Earth; the normal force, a support force perpendicular to a surface; tension, a pull transmitted by a taut cord; friction, a force between surfaces that opposes sliding or the tendency to slide; and an applied force, a push or pull from another object. Include only forces present in the situation. A normal force is not automatically equal to weight, and friction is not automatically present.

Name the system before drawing forces.
Show external forces acting on the chosen system.
Label each force arrow and its direction.

2. Use the net-force relationship

The net force is the vector sum of all external forces on a system. Opposite forces partly or fully cancel. Choose axes and a positive direction so each force component has a clear sign. A negative component means the force points opposite the chosen positive direction; it does not mean the force has a negative magnitude.

Newton’s second law relates net force, mass, and acceleration. Acceleration is a vector that describes how velocity changes. For the system as a whole, use its total mass and the net external force. Force is measured in newtons (

\mathrm{N}

), mass in kilograms (

\mathrm{kg}

), and acceleration in metres per second squared (

\mathrm{m/s^2}

).

For a group of objects moving together, forces between members are internal and are not included in the net external force on the whole group. They still matter if you isolate one member. A useful strategy is to find the group’s acceleration first, then analyse one object to find a contact force or tension.

\sum \vec{F}_{\mathrm{ext}}=m\vec{a}

Add force components with signs that match the chosen positive direction.
Zero net force means zero acceleration, not necessarily zero motion.
Use the whole system to find its acceleration; isolate an object to find an interaction force.

3. Predict, then investigate

A prediction uses a model: a stated description of the situation and its assumptions. For example, you might treat friction as negligible on a low-friction track if that assumption is suitable. Draw the force diagram, choose directions, list known values and the unknown, then apply the net-force relationship. Keep units in substitutions. Check units, sign, direction, significant figures, and whether the result is physically reasonable.

An investigation compares a prediction with measured evidence. One possible procedure is to connect a low-friction cart to a force sensor, record the cart’s mass, and measure applied force and acceleration. Repeat for several force settings while keeping the system and track conditions as consistent as possible. Before collecting data, predict how acceleration should change as applied force changes for the same total mass.

This is a proposed procedure, not a report of completed measurements. Record actual readings and measurement uncertainties only after carrying out the investigation. Compare observations with the prediction, allowing for measurement limits. If results differ, check the force diagram, mass, friction, sensor zero, and measurement method. Never alter recorded evidence to make it match a prediction.

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

Separate model-based predictions from measured evidence.
State what changes and what is kept constant.
Report only observations that were actually collected.

4. Keep the system and equation consistent

A force diagram and equation must describe the same system. If the system is one block, include the contact force from a neighbouring block when it acts on that block. If both blocks are the system, their contact interaction is internal and is left out of the external-force sum. Changing the system without changing the force list can lead to an incorrect result.

For motion in perpendicular directions, apply the force relationship separately in each direction. If an object stays at the same height, its vertical acceleration is zero, so vertical forces balance. Its horizontal forces need not balance. Analyse each direction using the stated reference frame and sign convention.

Check the result’s units and physical meaning. Dividing newtons by kilograms gives metres per second squared. A negative acceleration component points opposite the positive direction. It does not always mean the object’s speed is decreasing: direction of acceleration and direction of motion are separate.

1\ \mathrm{N}=1\ \mathrm{kg\,m/s^2}

Use one clear system boundary for each equation.
Balance forces only in directions with zero acceleration.
Interpret a negative sign as direction, not as negative magnitude.

Worked example

Two blocks pushed together

A

12.0\ \mathrm{kg}

block and an

8.0\ \mathrm{kg}

block touch on a level, low-friction surface. A horizontal

60.0\ \mathrm{N}

force pushes the

12.0\ \mathrm{kg}

block to the right. Predict the acceleration and the contact force on the

8.0\ \mathrm{kg}

block.

Set the system and direction
Use both blocks as the system and the room as the reference frame. Let right be positive. The total mass is $20.0\ \mathrm{kg}$ , and the applied force is $60.0\ \mathrm{N}$ to the right. The unknowns are the shared acceleration and the contact force.
Find the acceleration
For the two-block system, the contact forces between the blocks are internal: each block exerts a force on the other, and neither interaction is an external force on the pair. With friction treated as negligible, the only horizontal external force is $60.0\ \mathrm{N}$ to the right. Use the total mass in Newton’s second law.
$a=\frac{60.0\ \mathrm{N}}{12.0\ \mathrm{kg}+8.0\ \mathrm{kg}}=3.00\ \mathrm{m/s^2}$
Find the contact force
Now choose only the $8.0\ \mathrm{kg}$ block as the system. The contact force from the other block acts to the right and is its only horizontal force. It must produce the same acceleration as the two-block system.
$F_{\mathrm{contact}}=(8.0\ \mathrm{kg})(3.00\ \mathrm{m/s^2})=24.0\ \mathrm{N}$

Answer: The blocks accelerate at

3.00\ \mathrm{m/s^2}

to the right. The contact force on the

8.0\ \mathrm{kg}

block is

24.0\ \mathrm{N}

to the right.

Check: Newtons divided by kilograms gives metres per second squared, and kilograms times acceleration gives newtons. The contact force is less than the applied force because it accelerates only the lighter block.

Worked example

Two connected hanging masses

Two masses connected by a light cord pass over a pulley. One mass is

3.00\ \mathrm{kg}

and the other is

2.00\ \mathrm{kg}

. Treat the cord and pulley as ideal and ignore air resistance. Predict the acceleration and cord tension. Use

g=9.8\ \mathrm{m/s^2}

.

Choose systems and directions
Use both masses as the system to find acceleration, with the room as the reference frame. For the $3.00\ \mathrm{kg}$ mass, take down as positive. For the $2.00\ \mathrm{kg}$ mass, take up as positive. These choices match the masses’ linked motion along the cord. The cord is not part of the chosen system, so its tension forces on the masses are external forces.
Find the acceleration
The $3.00\ \mathrm{kg}$ mass moves down, while the $2.00\ \mathrm{kg}$ mass moves up. For the heavier mass, the positive direction is down: its weight contributes positively and tension negatively. For the lighter mass, the positive direction is up: tension contributes positively and weight negatively. When the two signed equations are added, the tension terms cancel: $-T+T=0$ . The weight terms leave a net downward drive equal to the difference between the weights.
$a=\frac{(3.00-2.00)\ \mathrm{kg}\,(9.8\ \mathrm{m/s^2})}{(3.00+2.00)\ \mathrm{kg}}=1.96\ \mathrm{m/s^2}$
Find the tension
Now isolate the $3.00\ \mathrm{kg}$ mass. Its weight points in the positive direction, and tension points opposite. The net force on it is its weight minus tension.
$T=(3.00\ \mathrm{kg})(9.8\ \mathrm{m/s^2})-(3.00\ \mathrm{kg})(1.96\ \mathrm{m/s^2})=23.5\ \mathrm{N}$

Answer: The acceleration is

1.96\ \mathrm{m/s^2}

, with the

3.00\ \mathrm{kg}

mass moving down and the

2.00\ \mathrm{kg}

mass moving up. The cord tension is

23.5\ \mathrm{N}

.

Check: The acceleration is in metres per second squared, and tension is in newtons. Tension is less than the heavier mass’s weight, consistent with that mass accelerating downward.

Worked example

A crate pulled across a floor

A

10.0\ \mathrm{kg}

crate is pulled horizontally to the right with a

50.0\ \mathrm{N}

force. The friction force on it is

15.0\ \mathrm{N}

to the left. Predict its acceleration.

Define the system and signs
The system is the crate, and the reference frame is the room. Let right be positive. The known horizontal forces are the applied pull and friction; the unknown is the horizontal acceleration.
Calculate the net force
The applied pull is positive and friction is negative because it points left. Add the signed horizontal force components.
$F_{\mathrm{net}}=50.0\ \mathrm{N}-15.0\ \mathrm{N}=35.0\ \mathrm{N}$
Apply the force relationship
Divide the net force by the crate’s mass. Since the result is positive, acceleration points right, in the direction of the net force.
$a=\frac{35.0\ \mathrm{N}}{10.0\ \mathrm{kg}}=3.50\ \mathrm{m/s^2}$

Answer: The crate’s acceleration is

3.50\ \mathrm{m/s^2}

to the right.

Check: The acceleration unit is metres per second squared. The direction is reasonable because the rightward pull exceeds the leftward friction force.

Common mistakes and how to avoid them

Including contact forces between members of a combined system in its external-force sum.

Correction: Those interactions are internal to the combined system. Include the relevant force when analysing just one member.

Assuming a moving object must have a net force in its direction of motion.

Correction: Net force determines acceleration. An object can move while its net force is zero.

Treating every force as positive.

Correction: Choose a positive direction and assign signs based on each force component’s direction.

Describing a proposed procedure as if it produced measured results.

Correction: Identify a procedure as proposed until measurements have actually been collected.

Lesson summary

Name the system, reference frame, and positive direction before solving.
Draw external forces acting on the chosen system; identify internal interactions.
Use net external force, mass, and acceleration consistently.
Check signs, direction, units, significant figures, and physical reasonableness.
Distinguish predictions from measurements in an investigation.

Check your understanding

Question 1

A

5.0\ \mathrm{kg}

cart has a net force of

10.0\ \mathrm{N}

to the left. What is its acceleration?

$2.0\ \mathrm{m/s^2}$ to the left
$2.0\ \mathrm{m/s^2}$ to the right
$50\ \mathrm{m/s^2}$ to the left
$0\ \mathrm{m/s^2}$

Show answer and explanation

2.0\ \mathrm{m/s^2}

to the left

Divide the net-force magnitude by mass:

10.0\ \mathrm{N}/5.0\ \mathrm{kg}=2.0\ \mathrm{m/s^2}

. Acceleration points in the direction of the net force, to the left.

Question 2

Two blocks are treated as one system. What should you do with the contact force one block exerts on the other?

Include it as an external force on the combined system.
Leave it out of the external-force sum because it is an internal interaction.
Replace it with the combined mass.
Set it equal to the system’s weight in every case.

Show answer and explanation

Leave it out of the external-force sum because it is an internal interaction.

Both blocks are inside the chosen system, so their contact interaction is internal. Include the contact force when analysing one block alone.

Question 3

A proposed cart investigation has not yet been performed. Which statement is appropriate?

The measured acceleration was a particular value.
The sensor proved the prediction correct.
The procedure can be used to collect force and acceleration measurements.
The simulated result is a completed physical measurement.

Show answer and explanation

The procedure can be used to collect force and acceleration measurements.

A proposed procedure describes a way to collect evidence. It does not provide measurements or establish an outcome.

Key terms

System: The object or group of objects selected for analysis.
Reference frame: The viewpoint used to describe position and motion.
External force: A force on the system from an object outside it.
Internal interaction: A force between objects that are both part of the chosen system.
Net force: The vector sum of external forces acting on a system.
Free-body diagram: A diagram representing a chosen object or system and showing the external forces acting on it.

Continue through SPH4U

View the complete SPH4U Ontario Grade 12 Physics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation B2.4. 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.

Official curriculum reference

Report a correction or ask a question

B2.4 · Predict and investigate forces acting on systems of objects

Ontario Grade 12 Physics

Ontario Grade 12 Physics · Study topic B2.4

What you will learn

1. Choose the system and map the forces

2. Use the net-force relationship

3. Predict, then investigate

4. Keep the system and equation consistent

Two blocks pushed together

Two connected hanging masses

A crate pulled across a floor

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SPH4U

About this lesson and its review