What Is Apparent Weight?

Your actual weight is the gravitational force acting on you.

It is calculated using:

Fg = mg

where:

• Fg = gravitational force or weight (N)
• m = mass (kg)
• g = gravitational field strength, approximately 9.8 N/kg near Earth's surface

Your actual weight does not suddenly change when an elevator starts moving.

However, how heavy you appear to be can change.

This is called apparent weight.

Apparent weight is the support force acting on an object, usually the normal force.

For a person standing on a scale:

Apparent weight = Normal force

or:

Wapparent = N

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Actual Weight vs Apparent Weight

Actual weight and apparent weight are not always the same.

Actual weight:

Fg = mg

It depends mainly on mass and gravitational field strength.

Apparent weight:

Wapparent = N

It depends on how strongly the supporting surface pushes on the object.

If you stand on a scale in an accelerating elevator, the scale measures the normal force, not the gravitational force directly.

That is why the scale reading can change even though your mass remains constant.

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Forces Acting on a Person in an Elevator

Consider a person standing on the floor of an elevator.

Two main forces act on the person:

↑ Normal force, N

● Person

↓ Weight, mg

The gravitational force always acts downward.

The elevator floor pushes upward on the person's feet. This upward contact force is the normal force.

The relationship between these forces determines the person's acceleration and apparent weight.

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Elevator at Rest

Suppose the elevator is stationary.

Acceleration is zero:

a = 0

Therefore:

Fnet = 0

The forces must balance:

N = mg

So:

Apparent weight = Actual weight

For example, consider a 60 kg person.

Actual weight:

Fg = mg

Fg = 60 × 9.8

Fg = 588 N

Therefore:

N = 588 N

The person's apparent weight is:

588 N

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Elevator Moving at Constant Velocity

Now suppose the elevator is moving upward at a constant velocity.

Because velocity is constant:

a = 0

Therefore:

Fnet = 0

and:

N = mg

The same is true if the elevator moves downward at constant velocity.

This is an important point:

Motion itself does not change apparent weight. Acceleration does.

An elevator can be moving quickly upward or downward, but if its velocity is constant:

N = mg

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Elevator Accelerating Upward

Suppose the elevator accelerates upward.

The person's acceleration is also upward.

Therefore, the upward force must be greater than the downward force:

N > mg

Taking upward as positive:

Fnet = N − mg

Using Newton's Second Law:

N − mg = ma

Therefore:

N = mg + ma

Factor out m:

N = m(g + a)

Since:

Apparent weight = N

we have:

Wapparent = m(g + a)

The person feels heavier.

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Why Do You Feel Heavier?

When an elevator accelerates upward, the floor must push upward on you strongly enough to accelerate you upward.

That means the normal force becomes greater than your weight.

Your body senses this increased contact force.

Therefore, you feel heavier.

Your actual gravitational weight has not increased.

Instead:

N has increased.

This is the source of the increased apparent weight.

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Worked Example 1: Accelerating Upward

A 70 kg person stands on a scale in an elevator accelerating upward at 2.0 m/s².

Calculate the apparent weight.

Use:

N = m(g + a)

N = 70(9.8 + 2.0)

N = 70(11.8)

N = 826 N

Therefore:

Apparent weight = 826 N

The person's actual weight is:

mg = 70 × 9.8

mg = 686 N

So the scale reads more than the person's actual weight.

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Elevator Accelerating Downward

Now suppose the elevator accelerates downward.

Weight is greater than the normal force:

mg > N

If downward acceleration has magnitude a:

mg − N = ma

Rearranging:

N = mg − ma

Therefore:

N = m(g − a)

So:

Wapparent = m(g − a)

The person feels lighter.

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Worked Example 2: Accelerating Downward

A 70 kg person is in an elevator accelerating downward at 2.0 m/s².

Calculate the apparent weight.

Use:

N = m(g − a)

N = 70(9.8 − 2.0)

N = 70(7.8)

N = 546 N

Therefore:

Apparent weight = 546 N

Actual weight remains:

686 N

The person therefore feels lighter.

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Comparing the Three Main Situations

For a person of mass m:

Therefore:

Upward acceleration → apparent weight increases

Zero acceleration → apparent weight equals actual weight

Downward acceleration → apparent weight decreases

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Direction of Motion Is Not Enough

A common mistake is to look only at whether the elevator is moving upward or downward.

What matters is the direction of acceleration, not simply the direction of velocity.

For example, an elevator can be:

moving upward but accelerating downward

This means the elevator is slowing down while moving upward.

In this situation:

N < mg

and the passenger feels lighter.

Similarly, an elevator can be:

moving downward but accelerating upward

This means it is slowing down while moving downward.

Then:

N > mg

and the passenger feels heavier.

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Velocity and Acceleration in Elevators

There are several possible combinations.

Moving upward and speeding up:

Acceleration upward → feel heavier

Moving upward and slowing down:

Acceleration downward → feel lighter

Moving downward and speeding up:

Acceleration downward → feel lighter

Moving downward and slowing down:

Acceleration upward → feel heavier

Moving at constant velocity:

Acceleration = 0 → normal apparent weight

The key question should always be:

Which direction is the acceleration?

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A Complete Elevator Journey

Consider a typical elevator traveling from a lower floor to a higher floor.

Stage 1: Starting upward

The elevator accelerates upward.

N > mg

Passenger feels heavier.

Stage 2: Moving upward at constant velocity

Acceleration = 0.

N = mg

Passenger feels normal weight.

Stage 3: Slowing near the upper floor

The elevator is still moving upward but accelerates downward.

N < mg

Passenger feels lighter.

Stage 4: Stopped

Acceleration = 0.

N = mg

Passenger feels normal weight again.

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Free-Body Diagrams for Elevator Problems

Free-body diagrams are extremely useful when solving apparent-weight problems.

For a passenger:

Accelerating upward:

↑↑ N

●

↓ mg

N > mg

---

No acceleration:

↑ N

●

↓ mg

N = mg

---

Accelerating downward:

↑ N

●

↓↓ mg

N < mg

The relative arrow lengths can be used to show which force is larger.

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Using Newton's Second Law

The safest method is not simply to memorize separate formulas.

Instead, begin with:

Fnet = ma

Choose a positive direction.

If upward is positive:

ΣFy = N − mg

Therefore:

N − mg = ma

Then use the sign of acceleration.

If acceleration is upward:

a is positive

If acceleration is downward:

a is negative

This single equation can solve all elevator apparent-weight situations.

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Worked Example 3: Using Signed Acceleration

A 50 kg passenger is in an elevator accelerating downward at 3.0 m/s².

Take upward as positive.

Therefore:

a = −3.0 m/s²

Use:

N − mg = ma

N − (50)(9.8) = 50(−3.0)

N − 490 = −150

N = 340 N

Apparent weight = 340 N

This method avoids needing separate equations for upward and downward acceleration.

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Calculating Acceleration from Apparent Weight

Sometimes the scale reading is given and the acceleration must be determined.

Suppose a 60 kg person stands on a scale reading 720 N.

Calculate the elevator's acceleration.

Actual weight:

mg = 60 × 9.8

mg = 588 N

Use:

N − mg = ma

720 − 588 = 60a

132 = 60a

a = 2.2 m/s²

Because N > mg, the net force is upward.

Therefore:

Acceleration = 2.2 m/s² upward

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Worked Example 4: A Lower Scale Reading

A 75 kg passenger stands on a scale reading 600 N.

Find the elevator's acceleration.

Weight:

mg = 75 × 9.8

mg = 735 N

Use:

N − mg = ma

600 − 735 = 75a

−135 = 75a

a = −1.8 m/s²

The negative sign means downward.

Therefore:

Acceleration = 1.8 m/s² downward

The passenger may actually be moving upward or downward—we cannot determine that from this information alone.

We only know the acceleration is downward.

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What Does a Scale Actually Measure?

A bathroom scale does not directly measure the gravitational force acting on you.

Instead, it responds to the contact force between you and the scale.

In an ordinary stationary situation:

N = mg

so the scale can convert the normal force into a mass reading.

For example, a scale may detect approximately 686 N and display:

70 kg

But if the scale is accelerating, the normal force changes.

The displayed value can therefore differ from the person's actual mass unless the scale accounts for the acceleration.

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Apparent Weight in Free Fall

Now consider an extreme situation.

Suppose the elevator and passenger are both falling freely with:

a = g

For downward acceleration:

N = m(g − a)

Therefore:

N = m(g − g)

N = 0

The apparent weight is zero.

The passenger is still affected by gravity.

Actual weight is still:

Fg = mg

But there is no supporting normal force.

Therefore:

Apparent weight = 0

This condition is called weightlessness or apparent weightlessness.

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Weightlessness Does Not Mean No Gravity

This is a very important distinction.

During free fall:

Gravity is still acting.

In fact, gravity is the force causing the person to accelerate downward.

The person feels weightless because there is no support force pushing against them.

Therefore:

Weightlessness does not mean Fg = 0.

It means:

N = 0.

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Apparent Weight Greater Than Normal

If an elevator accelerates upward strongly, the normal force can become significantly greater than the person's weight.

For example, suppose:

m = 80 kg

a = 4.0 m/s² upward

Then:

N = m(g + a)

N = 80(9.8 + 4.0)

N = 80(13.8)

N = 1104 N

Actual weight:

mg = 80 × 9.8

mg = 784 N

The person therefore experiences an apparent weight about 41% greater than normal.

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Apparent Weight as a Multiple of Normal Weight

Sometimes apparent weight is described using g-force.

If:

N = mg

the person experiences approximately:

1 g

If:

N = 2mg

the apparent weight is twice normal:

2 g

If:

N = 0

the person experiences apparent weightlessness:

0 g apparent weight

This idea becomes particularly important in:

• aircraft
• spacecraft
• roller coasters
• racing vehicles
• amusement rides

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Apparent Weight on Roller Coasters

The same physics occurs on amusement rides.

When the seat pushes strongly against you, your apparent weight increases.

You may feel pressed into the seat.

When the support force decreases, your apparent weight decreases.

You may feel lighter or feel as though you are lifting out of your seat.

Although circular-motion calculations can make roller coaster situations more complicated, the sensation of "heavier" or "lighter" is still closely related to changes in the normal force.

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Apparent Weight in Aircraft

Aircraft can also produce noticeable changes in apparent weight.

During certain maneuvers, passengers may experience:

• greater-than-normal apparent weight
• reduced apparent weight
• brief periods of apparent weightlessness

Aircraft used for weightlessness training follow special curved flight paths called parabolic flights.

During part of the maneuver, the aircraft and passengers are essentially falling together.

This produces a very small normal force and therefore apparent weightlessness.

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Astronauts and Apparent Weightlessness

Astronauts orbiting Earth appear weightless.

This is not because Earth's gravity has disappeared.

Gravity at the altitude of a low-Earth-orbit spacecraft is still substantial.

The spacecraft and astronauts are continuously falling toward Earth together while moving sideways fast enough to remain in orbit.

Because they fall together, there is very little normal support force between the astronaut and the spacecraft.

They therefore experience apparent weightlessness, commonly described as microgravity.

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Apparent Weight in a Car

You can even experience similar effects in a car.

When a car travels over a hill or through a dip, the normal force from the seat can change.

You may feel:

• lighter when passing over the top of a hill
• heavier when passing through a dip

Again, your actual gravitational weight has not suddenly changed.

The support force from the seat has changed.

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Apparent Weight and Mass

Mass does not change when apparent weight changes.

Suppose a person's mass is:

70 kg

Whether the person feels heavier, lighter, or weightless:

Mass = 70 kg

Their actual gravitational weight near Earth's surface remains approximately:

686 N

What changes is the normal force.

This distinction between mass, actual weight, and apparent weight is essential.

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Comparing Mass, Weight, and Apparent Weight

Mass is not a force.

Weight and apparent weight are forces.

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Elevator Cables

Elevator problems can also involve the tension in the cable supporting the elevator.

For an elevator car of mass M:

↑ T

●

↓ Mg

If the elevator accelerates upward:

T − Mg = Ma

Therefore:

T = M(g + a)

If the elevator accelerates downward:

T < Mg

If it moves at constant velocity:

T = Mg

The same Newton's Second Law reasoning applies to both passengers and the elevator itself.

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Worked Example 5: Elevator Cable Tension

An elevator has a total mass of 1200 kg and accelerates upward at 1.5 m/s².

Calculate the cable tension.

Use:

T − Mg = Ma

Therefore:

T = M(g + a)

T = 1200(9.8 + 1.5)

T = 1200(11.3)

T = 13,560 N

The cable tension is:

13.6 kN

approximately.

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Did You Know?

The brief "heavy" and "light" sensations you experience when an elevator starts or stops are direct evidence of Newton's Second Law.

You can investigate this using a scale in an elevator.

As the elevator begins accelerating upward, the reading increases.

During constant-speed motion, it returns to normal.

As the elevator slows near the upper floor, the reading decreases.

The entire sequence can be explained by changes in the normal force.

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Common Mistakes

Mistake 1: Saying actual weight changes in an elevator

Near Earth's surface, mg remains essentially constant. It is apparent weight that changes.

Mistake 2: Thinking upward motion means greater apparent weight

The important quantity is acceleration, not velocity.

Mistake 3: Thinking downward motion means lower apparent weight

An elevator moving downward but slowing has upward acceleration, so apparent weight is greater than normal.

Mistake 4: Saying weightlessness means no gravity

Apparent weightlessness occurs when the normal force becomes zero or nearly zero.

Mistake 5: Confusing mass with scale reading

Mass remains constant even when the support force changes.

Mistake 6: Forgetting the direction of acceleration

Always identify the acceleration before choosing the force equation.

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A Strategy for Solving Apparent-Weight Problems

1. Identify the person or object being analyzed.
2. Draw a free-body diagram.
3. Draw weight downward:

Fg = mg

4. Draw the normal force upward:

N

5. Determine the direction of acceleration.
6. Choose a positive direction.
7. Apply:

ΣF = ma

8. Solve for N.
9. Remember:

Apparent weight = N

10. Check whether the answer makes sense.

If acceleration is upward:

N > mg

If acceleration is zero:

N = mg

If acceleration is downward:

N < mg

If the object is in free fall:

N = 0

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Key Terms

Actual weight: The gravitational force acting on an object.

Apparent weight: The support force experienced by an object, usually the normal force.

Normal force: A contact force exerted perpendicular to a supporting surface.

Acceleration: The rate of change of velocity.

Free fall: Motion in which gravity is the only significant force acting.

Weightlessness: A condition in which apparent weight is zero or nearly zero.

g-force: A way of comparing apparent weight or acceleration effects with normal gravitational conditions.

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Key Equations

Actual weight:

Fg = mg

Apparent weight:

Wapparent = N

Using upward as positive:

N − mg = ma

Accelerating upward:

N = m(g + a)

Accelerating downward with acceleration magnitude a:

N = m(g − a)

No acceleration:

N = mg

Free fall:

N = 0

For the elevator car itself:

T − Mg = Ma

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Key Takeaways

• Apparent weight is the support force acting on an object, usually the normal force.
• A scale measures the normal force rather than gravitational weight directly.
• Actual weight is calculated using Fg = mg.
• Actual weight remains essentially constant during ordinary elevator motion.
• Apparent weight changes when the elevator accelerates.
• Upward acceleration produces N > mg, so you feel heavier.
• Downward acceleration produces N < mg, so you feel lighter.
• With zero acceleration, N = mg.
• The direction of velocity alone does not determine apparent weight.
• An elevator moving upward can produce either increased or decreased apparent weight depending on its acceleration.
• During free fall, N = 0, producing apparent weightlessness.
• Weightlessness does not mean that gravity has disappeared.
• Similar changes in apparent weight occur in elevators, cars, aircraft, spacecraft, and amusement rides.
• Free-body diagrams and Newton's Second Law provide a reliable way to solve apparent-weight problems.