In an accelerating vehicle, an apparent force called an inertial force acts. Master force balance and equations of motion in non-inertial frames.
Basic Knowledge
Inertial frame: A reference frame that is at rest or in uniform linear motion. Newton's laws hold directly.
Non-inertial frame: A frame moving with acceleration a₀. From within this frame, every object appears to experience an inertial force of mass × (−a₀).Inertial force = −ma₀ (a₀ is the acceleration vector of the frame)
Reading of a scale in an accelerating elevator:
・Accelerating upward at a: N = m(g + a) (apparently becomes heavier)
・Accelerating downward at a: N = m(g − a) (apparently becomes lighter)
・Free fall (a = g downward): N = 0 (weightless state)
When a train accelerates, a person inside observes an inertial force ma acting backward. This inertial force can be used to set up a balance equation (centrifugal force in a rotating frame is also a type of inertial force).
Magnitude of inertial force: ma₀ (m: mass of the object of interest, a₀: magnitude of the frame's acceleration)
Direction of inertial force: opposite to the frame's acceleration
Elevator accelerating upward: apparent weight = m(g + a)
Free fall: apparent weight = 0 (complete weightlessness)
Nature of inertial force: an apparent force (it does not actually exist)
Further Study (Background and Significance)
Inertial force is an apparent force, fundamentally different from real forces such as gravity, elastic force, and friction because it has no physical cause. However, it appears as a “felt force” to an observer in a non-inertial frame and is useful when solving balance and motion equations in that frame.
Einstein's equivalence principle states that “gravity and inertial force cannot be distinguished locally,” and it is the starting point of general relativity. An accelerating rocket in outer space can produce the same sensation as gravity.
The physical answer is the same regardless of whether the problem is solved in an inertial or non-inertial frame. In an inertial frame, use F = ma with only real forces; in a non-inertial frame, solve using real forces + inertial force = 0 (balance) or = ma' (relative acceleration).
Centrifugal force is the inertial force of a rotating frame, with magnitude mω²r and direction outward from the axis of rotation. It is also used only in a non-inertial frame (rotating frame).
- Inertial force = ma₀ (opposite to the frame's acceleration)
- Inertial frame: F = ma using only real forces
- Non-inertial frame: balance using real forces + inertial force
- Elevator accelerating upward: apparent weight increases
- Free fall: weightless state (N = 0)
- Centrifugal force is the inertial force of a rotating frame
- The answer is the same in either frame
Points to Note (Common Confusions)
① Do not mix inertial and non-inertial frames. State at the beginning which frame you will use. ② Inertial force does not actually exist. Do not add it to F = ma in an inertial frame. ③ During free fall, N = 0 (weightlessness), but gravity of mg is still acting (gravity has not disappeared). ④ If an elevator moves at constant speed, inertial force is zero (acceleration = 0).
Practice
- A person of mass 60 kg is in an elevator accelerating upward at 2.0 m/s². Find the scale reading (g = 10 m/s²).
- Find the angle θ at which a pendulum suspended from the ceiling tilts inside a train with acceleration a₀.
- State the criterion for deciding whether to solve a problem in an inertial or non-inertial frame.