B. Dynamics · Dynamics: Forces and Motion in a Plane

Inertial and Non-inertial Frames of Reference: Real and Fictitious Forces

B3.1 B2.1 A1.12About 70 minutes

Learning goals

I can tell the difference between an inertial frame (Newton's first law holds) and a non-inertial (accelerating) frame.
I can explain why observers in accelerating frames seem to feel fictitious forces, and why those forces have no real source.
I can analyse the same situation (an accelerometer in a car, a person in an elevator) from both frames and get the same physical result.
I can calculate apparent weight in an accelerating elevator and the acceleration shown by a pendulum accelerometer.

Words to know

inertial frame of reference
A frame that is not accelerating, in which Newton's first law holds: an object with no net force stays at rest or moves at a constant velocity.

non-inertial frame of reference
An accelerating frame (speeding up, slowing down or turning), such as a braking car or a spinning carousel.

fictitious (apparent) force
A force an observer in a non-inertial frame must add to make Newton's laws work. It has no source object and no third-law partner. Its size is ma(frame), opposite to the frame's acceleration.

centrifugal force
The fictitious outward force felt in a rotating frame.

apparent weight
The force a scale (or the floor) exerts to support you. It differs from mg when you are accelerating vertically.

Coriolis effect
In a rotating frame, moving objects appear to be deflected sideways. On Earth it steers large-scale winds and ocean currents.

Who pushed me?

You are standing on a city bus. It pulls away from the stop, and you lurch backward as if something shoved you. The air freshener hanging from the driver's mirror swings back too. But look around: nothing touched you. No hand, no rope, no wind.

From the sidewalk, the explanation is simple: the bus accelerated forward, and your body tended to stay where it was (inertia) until the floor and the handrail dragged you along. From inside the bus, it feels like a force pushed you backward. Both descriptions are consistent, once you understand frames of reference.

Inertial and non-inertial frames

A frame of reference is the point of view you measure from. An inertial frame is one in which Newton's first law works: objects with no net force on them keep a constant velocity. The ground is very nearly an inertial frame for everyday motion (Earth's rotation and orbit cause only small effects). So is anything moving at a constant velocity relative to the ground: on a perfectly smooth train at constant speed, a ball on the floor stays put.

A non-inertial frame is accelerating: a car speeding up, braking or turning; an elevator starting or stopping; a spinning merry-go-round. In such a frame, objects seem to accelerate with no force acting on them. To use F = ma inside it, the observer must add a fictitious force:

F(fictitious) = −m a(frame)

It points opposite to the frame's acceleration. It is called fictitious because no object exerts it and it has no third-law partner. Real forces (gravity, normal, tension, friction) are the same in every frame.

Worked example

Example 1: The pendulum accelerometer

An air freshener hangs on a string from the mirror of a car that is accelerating forward at 2.0 m/s². At what angle to the vertical does the string hang? Solve in both frames.

Ground frame (inertial): only two real forces act: tension along the string and gravity. Their sum must equal ma, forward. Vertical: FT cos θ = mg. Horizontal: FT sin θ = ma.
Divide: tan θ = a ÷ g = 2.0 ÷ 9.8, so θ = 11.5°, with the string tilted backward (the bob trails behind the mirror).
Car frame (non-inertial): the bob is at rest, so the forces must balance. Tension and gravity alone cannot balance, so add a fictitious force ma pointing backward. Horizontal: FT sin θ = ma. Vertical: FT cos θ = mg. Same equations, same answer.
Turned around, this makes a measuring tool: a = g tan θ.

11.5° from the vertical, tilted backward. Both observers agree on the angle; they disagree only on how to describe why.

The same hanging bob seen from two frames. In (a), the real forces add to a forward net force, ma, so the bob accelerates with the car. In (b), the bob is at rest relative to the car, so the passenger adds a fictitious force −ma (backward) to make the forces balance.

Worked example

Example 2: Apparent weight in an elevator

A 60 kg student stands on a bathroom scale in an elevator. What does the scale read when the elevator (a) speeds up going up at 1.5 m/s², (b) moves at a constant velocity, (c) speeds up going down at 1.5 m/s², and (d) falls freely?

The scale reading is the normal force FN. Up is +. Newton's second law: FN − mg = ma, so FN = m(g + a).
(a) a = +1.5: FN = 60(9.8 + 1.5) = 678 N.
(b) a = 0: FN = 60 × 9.8 = 588 N.
(c) a = −1.5: FN = 60(9.8 − 1.5) = 498 N.
(d) a = −9.8: FN = 0.

(a) 678 N (b) 588 N (c) 498 N (d) 0 N. Your true weight (mg) never changes; your apparent weight does. Slowing down while going down feels the same as speeding up while going up: both are an upward acceleration.

Scale readings for the 60 kg student in Example 2. In the elevator's frame, a fictitious force −ma adds to or subtracts from gravity, so the student feels heavier or lighter. In free fall, the scale reads zero: this is the "weightlessness" astronauts feel in orbit.

Rotating frames: centrifugal and Coriolis

On a spinning carousel, you feel thrown outward. In the carousel's rotating frame, that feeling is the centrifugal force, a fictitious force pointing away from the centre. In the ground frame there is no outward force at all: your body tries to go straight (first law), and the bar you hold pulls you inward to keep you on the circle. You will analyse that inward force in Centripetal Force.

A rotating frame has a second fictitious force, the Coriolis effect: objects moving across a spinning surface seem to curve sideways. Earth is a slowly rotating frame, so the Coriolis effect is tiny for a thrown ball but important over thousands of kilometres. It deflects large-scale winds to the right in the Northern Hemisphere, which is why the winds circulating around a low-pressure system such as a hurricane turn counterclockwise north of the equator. (The popular claim that sinks drain the opposite way in the two hemispheres is a myth: the effect is far too weak at that size.)

Real or fictitious?

Fictitious forces are not "fake" feelings: your body really is pressed into the seat. What is fictitious is the explanation "a force pushed me". In the ground frame, the seat pushes you forward. A good test: ask "what object exerts this force?" If there is no answer, it is fictitious.

Just for fun

Why do fictitious forces never win an argument?

They have no real basis, and when you ask who sent them, they cannot name anyone.

Hands-on activity

Accelerometers you can make

Do these only as a passenger, never while driving, and keep any device secured so it cannot fly around.

1. Pendulum accelerometer. Tape a protractor to a piece of cardboard with its straight edge on top. Tie a thread to the centre hole and hang a metal washer or an eraser from it. Hold the card level and pointing forward in a car or bus. When the vehicle speeds up, read the angle θ from the vertical and calculate a = 9.8 tan θ. Compare speeding up, braking and turning.

2. Bubble accelerometer. Fill a clear water bottle almost to the top, leaving a small air bubble, and lay it on its side, pointing forward. Watch the bubble when the vehicle speeds up and brakes. (Surprise: the bubble moves forward when you speed up, because the water, being denser, "lags" backward and squeezes the bubble forward.)

3. Elevator scale. With a grown-up, stand on a bathroom scale in an elevator. Record the readings as it starts, moves steadily and stops, going up and going down. Use FN = m(g + a) to find each acceleration.

Your turn

A pendulum in a car hangs at 8.0° from the vertical, tilted backward. What is the car's acceleration?

A 50 kg person on a scale in an elevator sees a reading of 560 N. What is the elevator's acceleration? Describe two motions that could produce it.

Is a car going around a curve at a constant speed an inertial frame? Explain.

When a car brakes hard, passengers lurch forward. Name the fictitious force in the car's frame and the real force that stops the passengers.

What does a bathroom scale read in an elevator whose cable has snapped (ignore air resistance and safety brakes)?

Which way do the winds around a hurricane circulate in the Northern Hemisphere, and which fictitious effect causes it?

More practice: warm-up

In which kind of frame does Newton's first law hold?

Does a fictitious force have a third-law partner?

Is a train moving at a constant 100 km/h on a straight, smooth track an inertial frame (approximately)?

In which direction does a fictitious force point?

Does your true weight, mg, change in an accelerating elevator?

More practice: on your own

A car accelerates forward at 3.0 m/s². At what angle does a pendulum accelerometer hang?

A 70 kg person rides an elevator. Find the scale reading when it speeds up going down at 2.0 m/s², and when it moves down at constant speed.

A coffee cup sits on a car's dashboard with μs = 0.30. What is the largest braking deceleration before it slides? Explain in the car's frame.

A 2.0 kg mass hangs from a spring scale in an elevator. The scale reads 22 N. Find the elevator's acceleration.

A child on a merry-go-round says, "A force is pushing me outward." Explain what is really happening in the ground frame.

Challenge

A passenger on a train accelerating forward at 1.2 m/s² drops a ball from 1.5 m above the floor. Where does it land relative to the point directly below where it was released? Explain in both frames.

A 60 kg student rides an elevator up from rest. The scale reads 700 N for 2.0 s, then 588 N for 5.0 s, then 500 N until the elevator stops. Find the maximum speed and the total distance travelled.

A helium balloon on a string is held inside a closed car. When the car accelerates forward, the balloon tilts forward, the opposite of the hanging air freshener. Explain.

Check your understanding

Quiz

Which of these is an inertial frame of reference (to a good approximation)?

A fictitious force in an accelerating frame:

A 50 kg person stands on a scale in an elevator that is slowing down while moving up at 2.0 m/s². The scale reads:

A pendulum accelerometer in a car hangs at 45° from the vertical. The car's acceleration is:

In the ground frame, what keeps a rider on a carousel moving in a circle?

A person's apparent weight is zero when:

For parents and teachers

Big idea: Newton's laws hold in inertial (non-accelerating) frames. In accelerating frames, observers must add fictitious forces, which have no source (B3.1). Analysing the same situation from both frames builds a deep understanding of inertia.

How to help: On a car ride (as passengers), ask your teen to predict which way a hanging object will swing when you speed up, brake and turn, and to explain it both "from the sidewalk" and "from the seat".

Common difficulties:

Treating centrifugal force as a real force in the ground frame.
Mixing frames in one free-body diagram (adding a fictitious force while also writing ΣF = ma for the acceleration).
Thinking the scale reading changes because gravity changes.
Confusing "moving down" with "accelerating down".

Safety: do the accelerometer activities only as a passenger, with objects secured.