Pseudo Force

The problem with non-inertial frames

Newton's laws hold in inertial frames — frames that are at rest or moving at constant velocity. In an accelerating frame (a car braking, a rocket launching, a spinning merry-go-round), Newton's laws appear to break down.

In an accelerating frame, objects seem to experience forces with no visible cause. A passenger in a braking car is thrown forward — but no one is pushing them. These apparent forces are called pseudo forces (also called fictitious forces or inertial forces).

The pseudo force

If a reference frame accelerates at $\vec{a}_{frame}$ relative to an inertial frame, then in this non-inertial frame, every object of mass $m$ appears to experience an additional force:

$\vec{F}_{pseudo} = -m\vec{a}_{frame}$

This force is opposite to the acceleration of the frame.

Derivation

In an inertial frame, Newton's Second Law holds:

$\vec{F}_{real} = m\vec{a}_{inertial}$

where $\vec{a}_{inertial}$ is the acceleration in the inertial frame.

The acceleration in the non-inertial frame:

$\vec{a}_{non-inertial} = \vec{a}_{inertial} - \vec{a}_{frame}$

Substituting:

$\vec{F}_{real} = m(\vec{a}_{non-inertial} + \vec{a}_{frame})$

$\vec{F}_{real} - m\vec{a}_{frame} = m\vec{a}_{non-inertial}$

$\vec{F}_{real} + \underbrace{(-m\vec{a}_{frame})}_{\vec{F}_{pseudo}} = m\vec{a}_{non-inertial}$

In the non-inertial frame, Newton's Second Law can be "restored" by adding $\vec{F}_{pseudo}$:

$\vec{F}_{real} + \vec{F}_{pseudo} = m\vec{a}_{non-inertial}$

Examples

Car accelerating forward:

Frame accelerates at $+a$ (forward). Pseudo force on passenger: $-ma$ (backward). Passenger feels pushed back into the seat.

Car braking:

Frame decelerates: $\vec{a}_{frame} = -a$ (backward). Pseudo force: $+ma$ (forward). Passenger feels thrown forward.

Elevator accelerating upward:

$\vec{a}_{frame}$ upward. Pseudo force downward: $F_{pseudo} = ma$ downward. Combined with gravity, effective gravity increases: $g_{eff} = g + a$.

Rotating frame:

In a rotating frame, two pseudo forces arise: centrifugal force (outward, $= m\omega^2 r$) and Coriolis force (perpendicular to velocity). These explain trade winds, ocean currents, and the rotation of cyclones.

When to use pseudo forces

Approach 1 (inertial frame): Apply real forces only. Use the ground (inertial frame) as reference. Generally preferred and less error-prone.

Approach 2 (non-inertial frame): Add pseudo force $-m\vec{a}_{frame}$ to real forces. Treat the non-inertial frame as if it were inertial. Useful when the problem is described from the accelerating frame's perspective.

Both approaches give the same physical answers.