Apparent Weight (College Board AP® Physics 1: Algebra-Based)

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Apparent weight

Apparent weight and normal force

The magnitude of the apparent weight of a system is the magnitude of the normal force exerted on the system
- The apparent weight of an object is the weight it appears to be and not the weight it actually is

$apparent weight = |\vec{F_{n}}|$

The normal force is the perpendicular component of the force exerted on an object by the surface with which it is in contact
- The normal force is a contact force acting against the weight of an object

Normal contact force

An object with mass "m" has an upward reaction force and downward weight. Caption reads "Reaction forces support an object by balancing its weight." — **An object resting on another in contact has a normal reaction force acting in the opposite direction to its weight**

Apparent weight can be observed when an object is immersed in a fluid (liquid or gas)
- The magnitude of the normal force is equal to the buoyant force acting upwards on the object
- The buoyant force is equal to the weight of the fluid displaced by the object

$apparent weight = |{\vec{F}}_{n}| = buoyant force = weight of fluid displaced$

Buoyant force

Apparent weight and gravitational force

If a system is accelerating:
- the apparent weight of the system is not equal to the magnitude of the gravitational force exerted on the system
- the apparent weight of a system can be greater than or less than the actual weight, depending on the direction of acceleration

When the system is stationary:
- for example, an object placed on a bathroom scale
- the reading on the scale will equal the object's actual weight

$apparent weight = |\vec{F_{n}}| = actual weight = m g$

When the system is accelerating:
- for example, if the scale is placed in an elevator that is accelerating
- the reading on the scale will be equal to the object's apparent weight
When the elevator is accelerating positively (speeding up):
- the normal force exerted by the scale on the object is greater than object's actual weight
- the scale exerts an upward force equal to the object's actual weight plus the additional force producing the acceleration

$apparent weight = |\vec{F_{n}}| = m g + m a$

When the elevator is accelerating negatively (slowing down):
- the normal force exerted by the scale on the object is less than the object's actual weight
- the scale exerts an upward force equal to the object's actual weight minus the additional force producing the acceleration

$apparent weight = |\vec{F_{n}}| = m g - m a$

Apparent weight of an object accelerating in different directions

Three diagrams show the normal force on an object: stationary (Fn=mg), positive acceleration (Fn=mg+ma), and negative acceleration (Fn=mg-ma). — **The magnitude of the apparent weight of a person in an elevator changes depending on the direction of the acceleration**

Gravitational force equivalence principle

The equivalence principle states that an observer in a non-inertial reference frame is unable to distinguish between an object’s apparent weight and the gravitational force exerted on the object by a gravitational field
- A non-inertial reference frame is a reference frame that is accelerating with respect to an inertial reference frame

Moving in a non-inertial reference frame

An example of an object moving in a non-inertial reference frame is when an external net force is applied to accelerate or decelerate a vehicle
- The passengers and the car no longer move together
- The inertia of the passengers causes them to resist a change in motion as the car accelerates and moves differently to them

When a vehicle accelerates, an object inside will initially move backwards as it resists the change in motion
When a vehicle decelerates, an object in the vehicle will continue to move forward as it resists the change in motion

Resisting the change in motion of a moving vehicle

Two images depict a car with a box on its roof. As the car moves forward, the box moves backward. When the car stops, the box moves forward and falls off. — **When a car starts to move a box on the top resists the change in motion and moves backwards in the opposite direction.**

The equivalence principle implies that experiencing a constant acceleration, $a$ , not in a gravitational field is equivalent to being at rest in a uniform gravitational field with gravitational field strength, $g$ where $a = g$
For example, if an elevator is in space beyond the reach of a gravitational field and the elevator experiences a constant acceleration of $10 m / s^{2}$ then:
- objects in the elevator fall with the same magnitude of constant acceleration of $a = 10 m / s^{2}$
- a person inside the elevator can not tell the difference between being accelerated inside the elevator and being stationary on Earth where $g = 10 m / s^{2}$

An elevator accelerating in space

Elevator accelerating upwards at 10 m/s². A ball inside is dropped, also accelerating at 10 m/s² downwards, meeting the elevator. Text explains the scenario. — **An object dropped in an elevator that is accelerating at 10 m/s^2 away from any gravitational field will also drop at 10 m/s^2**

An elevator stationary on Earth

A stick figure holding a purple ball in a stationary elevator on Earth is shown above an illustration of the Earth. Text reads, "a = g = 10 m/s²." — **The person in the elevator cannot tell the difference between being stationary on Earth and accelerating at 10 m/s^2 far away from any gravitational field**

It can be shown algebraically in a non-inertial reference frame that an object’s apparent weight and the gravitational force exerted upon it by a gravitational field appear equal to an observer also in the non-inertial reference frame
From the reference frame of the accelerating elevator, the apparent weight of the person is

$|{\vec{F}}_{n}| = m a$

From the reference frame of the stationary person in the stationary elevator on Earth, their apparent weight is equal to their actual weight

$|{\vec{F}}_{n}| = |{\vec{F}}_{g}| = m g$

The two scenarios are equivalent
- The laws of Physics will work in exactly the same way for both

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