GRE Physics Companion: Weight and Normal Force

This companion is designed for rapid review after M02-04. The main strategy is to stop treating N = mg as a memorized rule. Instead, choose the direction Normal to the contact or inward along a curved path and write Newton’s second law in that direction.

1 Fast triage

For a vertical support or elevator, choose a vertical sign convention and write

(1)

For an incline, choose a normal axis and write

(2)

For a curved path, inward acceleration has magnitude

(3)

so use

(4)

2 High value traps

A scale measures the support force, not mg directly.

Free fall can give

(5)

even while gravity remains strong.

On a hill crest, inward is downward, so

(6)

At the bottom of a valley, inward is upward, so

(7)

If a maintained contact calculation gives N < 0, the body has already lost contact. Use N = 0 at the threshold.

3 Worked GRE example 1: moving upward but slowing down

An elevator passenger is moving upward but slowing down with acceleration magnitude a. Is the scale reading greater than, equal to, or less than mg?

Moving upward while slowing means the acceleration points downward. Taking upward as positive,

(8)

Thus

(9)

so

(10)

Velocity direction is irrelevant to this instantaneous force balance.

4 Worked GRE example 2: hill contact threshold

A vehicle moves over the crest of a hill of radius R. At what speed does it just lose contact?

At the crest,

(11)

At the threshold of contact loss,

(12)

Therefore

(13)

and

(14)

The mass cancels.

5 GRE speed questions

1. A person stands on a scale in an elevator accelerating upward. The scale reading is (A) less than mg (B) equal to mg (C) greater than mg (D) zero.
2. An elevator moves downward at constant velocity. The scale reading is (A) zero (B) less than mg (C) equal to mg (D) greater than mg.
3. A block rests on a frictionless incline of angle 𝜃 with no acceleration normal to the plane. The normal force is (A) mg (B) mg sin 𝜃 (C) mg cos 𝜃 (D) mg tan 𝜃.
4. A CAR passes over the crest of a circular hill at nonzero speed. Compared with mg, the normal force is generally (A) smaller (B) equal (C) larger (D) unrelated to the motion.
5. A rider passes through the bottom of a circular dip. Compared with mg, the support force is generally (A) smaller (B) equal (C) larger (D) zero.
6. A person and scale are both in ideal free fall. Which statement is correct? (A) N = 0 and gravity is zero (B) N = 0 but gravity is not zero (C) N = mg (D) N > mg.

6 Answers and rationales

1. C. Upward acceleration requires N − mg = ma > 0.
2. C. Constant velocity means zero acceleration, so N = mg.
3. C. The normal component of Weight is mg cos 𝜃.
4. A. At the crest, mg − N = mv²∕R, so N < mg.
5. C. At the bottom, N − mg = mv²∕R, so N > mg.
6. B. Free fall removes the support force, not the gravitational force.

References