B3.1 Distinguish inertial and non-inertial frames and apparent forces

Ontario Grade 12 Physics

Dynamics

Recognizing apparent forces in different frames of reference

A passenger in a bus may feel pushed backward when the bus speeds up. A person watching from the sidewalk does not see a backward force acting on the passenger; the passenger’s body tends to keep its motion while the bus moves forward. The difference comes from the frame used to describe the motion. This lesson examines inertial and non-inertial frames and the apparent forces used in a non-inertial frame. An apparent force helps make Newton’s laws usable in that accelerating frame. It is not an extra interaction, such as a push or a pull from another object.

What you will learn

Define a physical system and a reference frame, and choose a positive direction.
Distinguish inertial frames from non-inertial frames.
Explain why an apparent force is used in an accelerating frame.
Compare descriptions of the same motion from different frames.

1. Prerequisite bridge: systems, frames, and Newton’s laws

A physical system is the object or group of objects being studied. In an example about a passenger, the passenger can be the system. A reference frame is the viewpoint used to describe position and motion. A frame includes a coordinate system and an observer who measures motion relative to it.

Before solving a motion problem, state the system, frame, and positive direction. For example, the system may be a ball, the frame may be the ground, and forward may be positive. Position, velocity, and acceleration are vectors: each has magnitude and direction. Mass and time are scalars: each has magnitude but no direction.

In SPH3U mechanics, Newton’s second law relates the net force on a system to its mass and acceleration. The net force is the vector sum of the real forces acting on the system. In SI units, force is measured in newtons, mass in kilograms, and acceleration in metres per second squared.

\sum \vec{F}=m\vec{a}

State the system, frame, and positive direction before interpreting signs.
A negative component means the vector points opposite to the chosen positive direction.
Newton’s second law connects net force and acceleration in an inertial frame.

2. Inertial and non-inertial frames

An inertial frame is a frame that is not accelerating. A frame at rest relative to the ground can often be treated as inertial for ordinary classroom motion problems. A frame moving at constant velocity relative to an inertial frame is also inertial. In such a frame, an object with zero net force remains at rest or continues at constant velocity.

A non-inertial frame is accelerating relative to an inertial frame. The frame might speed up, slow down, or change direction. A passenger’s frame inside an accelerating car is non-inertial. The car’s frame changes its velocity, even if the passenger is sitting still relative to the seat.

Newton’s second law, used with only real forces, does not directly describe motion in a non-inertial frame in the usual way. To use the familiar force-and-acceleration relationship within that frame, include an apparent force. An apparent force is a force-like term that accounts for the acceleration of the frame. It does not come from a separate object interacting with the system.

For a frame that accelerates in a straight line, the apparent force on a system points opposite to the frame’s acceleration. Its size depends on the system’s mass and the frame’s acceleration. The frame acceleration is measured relative to an inertial frame. The apparent force is a vector and is measured in newtons.

\vec{F}_{\mathrm{app}}=-m\vec{a}_{\mathrm{frame}}

Inertial frames have no acceleration relative to one another.
A non-inertial frame accelerates relative to an inertial frame.
In a translating non-inertial frame, apparent force points opposite the frame’s acceleration.

3. Using apparent forces to describe motion

There are two consistent ways to describe a situation. In an inertial frame, draw the real forces on the system and use Newton’s second law. In a non-inertial frame, include the apparent force as well as the real forces. Do not add an apparent force to an inertial-frame force diagram.

For a car accelerating forward, a passenger viewed from the ground accelerates forward with the car. A real force from the seat or seat belt can provide that forward acceleration. From the passenger’s frame, the passenger is at rest relative to the car. The apparent force points backward, opposite the car’s acceleration, and balances the forward real force in that frame.

A turn provides another useful comparison. In the ground frame, an object moving in a circular path must have a net real force toward the centre of the path. In a frame turning with the vehicle, an apparent force points outward, away from the centre. This outward apparent force is included only when describing the motion in the turning frame.

The system and frame must be identified before drawing forces. A force diagram for the ground frame and one for the turning frame can differ because the second frame is accelerating. The difference does not mean the object has gained a new physical interaction.

\sum \vec{F}_{\mathrm{real}}+\vec{F}_{\mathrm{app}}=m\vec{a}_{\mathrm{frame\ view}}

Real forces are caused by interactions with other objects or fields.
Apparent forces are included to describe motion from a non-inertial frame.
Check the frame before deciding which forces belong on a diagram.

4. A reliable comparison method

First identify the system and choose a frame. Mark the positive direction. Next decide whether the frame accelerates relative to an inertial frame. If it does not, describe the motion using real forces alone. If it does, either switch to an inertial frame or include the appropriate apparent force in the non-inertial frame.

Keep the two descriptions separate. In the inertial frame, use the system’s actual acceleration and sum the real forces. In the non-inertial frame, use the system’s acceleration relative to that frame and include apparent force. A sign check helps: if the frame accelerates forward, its apparent force points backward.

A calculated force should have units of newtons. Its direction should agree with the chosen sign convention and the frame’s acceleration. Check that the result makes sense: a larger system mass or a larger frame acceleration gives a larger apparent-force magnitude.

F_{\mathrm{app}}=m a_{\mathrm{frame}}

A frame’s acceleration determines the direction of the apparent force.
Do not confuse acceleration relative to the frame with acceleration relative to the ground.
Check units, sign, direction, and physical reasonableness.

Worked example

A passenger in a forward-accelerating bus

A bus accelerates forward at

1.8\ \mathrm{m/s^2}

. A passenger has a mass of

60.\ \mathrm{kg}

. Find the passenger’s apparent force in the bus frame. Let forward be positive.

Set the frame and system
The system is the passenger. The bus frame accelerates forward relative to the ground, so it is non-inertial. Forward is positive, and the bus-frame acceleration is $+1.8\ \mathrm{m/s^2}$ .
Apply the apparent-force relationship
The apparent force is opposite the frame’s acceleration. Use the passenger’s mass and the bus-frame acceleration.
$F_{\mathrm{app}}=-m a_{\mathrm{frame}}$
Substitute and interpret
The result is negative, so the apparent force points backward, opposite the chosen positive direction.
$F_{\mathrm{app}}=-(60.\ \mathrm{kg})(1.8\ \mathrm{m/s^2})=-1.1\times10^2\ \mathrm{N}$

Answer: The passenger’s apparent force in the bus frame is

1.1\times10^2\ \mathrm{N}

backward.

Check: The units are

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

. The negative sign means backward. The magnitude is reasonable for a

60.\ \mathrm{kg}

passenger in a frame accelerating at

1.8\ \mathrm{m/s^2}

.

Worked example

Comparing the bus and ground descriptions

A bus and passenger accelerate forward at

2.0\ \mathrm{m/s^2}

. The passenger’s mass is

55\ \mathrm{kg}

. In the ground frame, find the net real force on the passenger. Then state the apparent force in the bus frame. Take forward as positive.

Use the inertial-frame description
The system is the passenger, and the ground frame is treated as inertial. In this frame, Newton’s second law gives the net real force from the passenger’s forward acceleration.
$\sum F_{\mathrm{real}}=(55\ \mathrm{kg})(2.0\ \mathrm{m/s^2})=1.1\times10^2\ \mathrm{N}$
Use the bus-frame description
The passenger is at rest relative to the bus, but the bus frame accelerates forward. The apparent force in that frame is backward and has the same magnitude as the forward net real force in this situation.
$F_{\mathrm{app}}=-(55\ \mathrm{kg})(2.0\ \mathrm{m/s^2})=-1.1\times10^2\ \mathrm{N}$

Answer: In the ground frame, the net real force is

1.1\times10^2\ \mathrm{N}

forward. In the bus frame, the apparent force is

1.1\times10^2\ \mathrm{N}

backward.

Check: Both results have force units. Their opposite directions fit the frame comparison: the passenger accelerates forward relative to the ground but is at rest relative to the accelerating bus.

Worked example

An object described from a turning frame

A

0.40\ \mathrm{kg}

object moves with a vehicle around a curve. At one instant, the vehicle’s turning frame has a centre-directed acceleration of

3.0\ \mathrm{m/s^2}

. Find the apparent-force direction and magnitude in that frame.

Define the system and direction
The system is the object. The frame is attached to the turning vehicle, so it accelerates toward the curve’s centre. Choose the centre direction as positive.
Find the apparent force
The apparent force points opposite the frame’s centre-directed acceleration. Its magnitude is the product of the mass and acceleration magnitude.
$F_{\mathrm{app}}=-(0.40\ \mathrm{kg})(3.0\ \mathrm{m/s^2})=-1.2\ \mathrm{N}$

Answer: The apparent force is

1.2\ \mathrm{N}

outward, away from the curve’s centre.

Check: The negative sign indicates the outward direction under the chosen convention. The units reduce to newtons, and the force is modest for a

0.40\ \mathrm{kg}

object at the stated acceleration.

Common mistakes and how to avoid them

Calling every frame that is moving an accelerating frame.

Correction: A frame moving at constant velocity relative to an inertial frame is also inertial. Acceleration, not motion alone, makes a frame non-inertial.

Treating apparent force as a real push from the seat, road, or another object.

Correction: A real force comes from an interaction. Apparent force accounts for the acceleration of the chosen non-inertial frame.

Drawing an apparent force in a ground-frame diagram when the ground frame is treated as inertial.

Correction: Use real forces alone in that inertial frame. Include apparent force only when describing motion in a non-inertial frame.

Pointing the apparent force in the same direction as the frame’s acceleration.

Correction: For a translating frame, apparent force points opposite the frame’s acceleration.

Lesson summary

An inertial frame is not accelerating relative to an inertial frame; a non-inertial frame is accelerating.
In an inertial frame, relate real net force to the system’s acceleration.
In a non-inertial frame, include apparent force opposite the frame’s acceleration.
Always name the system, frame, and positive direction, then check units and direction.

Check your understanding

Question 1

A vehicle accelerates to the right. In the vehicle’s frame, which way does the apparent force on a passenger point?

To the right
To the left
It points upward
It is always zero

Show answer and explanation

To the left

The vehicle frame accelerates to the right, so the apparent force points in the opposite direction: left.

Question 2

A frame moves at constant velocity relative to an inertial frame. How should it be classified?

Non-inertial, because it is moving
Inertial, because it is not accelerating
Non-inertial, because its velocity is non-zero
It cannot be classified without knowing its mass

Show answer and explanation

Inertial, because it is not accelerating

A frame moving at constant velocity has no acceleration relative to the inertial frame, so it is also inertial.

Question 3

A

2.0\ \mathrm{kg}

system is viewed from a frame accelerating forward at

3.0\ \mathrm{m/s^2}

. What is its apparent force in that frame?

$6.0\ \mathrm{N}$ forward
$6.0\ \mathrm{N}$ backward
$1.5\ \mathrm{N}$ backward
$0\ \mathrm{N}$

Show answer and explanation

6.0\ \mathrm{N}

backward

The magnitude is the mass times the frame-acceleration magnitude,

6.0\ \mathrm{N}

. Its direction is opposite the frame acceleration, so it points backward.

Key terms

Physical system: The object or group of objects chosen for analysis.
Reference frame: The viewpoint and coordinate system used to describe position and motion.
Inertial frame: A frame that is not accelerating relative to an inertial frame.
Non-inertial frame: A frame that is accelerating relative to an inertial frame.
Apparent force: A force-like term included to describe motion in an accelerating frame; it does not represent a separate interaction.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation B3.1. 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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B3.1 · Distinguish inertial and non-inertial frames and apparent forces

Ontario Grade 12 Physics

Recognizing apparent forces in different frames of reference

What you will learn

1. Prerequisite bridge: systems, frames, and Newton’s laws

2. Inertial and non-inertial frames

3. Using apparent forces to describe motion

4. A reliable comparison method

A passenger in a forward-accelerating bus

Comparing the bus and ground descriptions

An object described from a turning frame

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