A spring balance measures the tension T in the spring, which equals the apparent weight of the object hanging from it. In an elevator: free body diagram for mass m: T (upward from spring) and mg (downward from gravity). Newton's second law (taking upward positive): T − mg = ma_net. So T = m(g + a_net). Sign convention: a_net is POSITIVE if the net acceleration is upward, NEGATIVE if downward. (1) At rest or constant velocity: a_net = 0 → T = mg (reads true weight). (2) Accelerating upward at a: a_net = +a → T = m(g+a) (reads MORE — feels heavier). (3) Accelerating downward at a: a_net = −a → T = m(g−a) (reads LESS — feels lighter). (4) Free fall: a_net = −g → T = m(g−g) = 0 (reads zero — weightlessness).

• Apparent weight = T = m × g_eff where g_eff = (g + a_net). If elevator decelerates while going up (deceleration = a downward), a_net = −a → T = m(g−a) < mg — misleadingly, decelerating upward also reads light. If elevator decelerates while going down (deceleration = a upward), a_net = +a → T = m(g+a) > mg. Common NEET trap: a person in an elevator going DOWN and DECELERATING feels HEAVIER (T > mg) because deceleration while going down = upward acceleration. Read 'direction of acceleration' not 'direction of motion'.
• Spring balance on the Moon: g_moon ≈ g/6. T = mg_moon = mg/6. The spring balance reads 1/6 of Earth value. The object TRULY hasn't changed (same mass) but the spring reads less because gravitational pull is weaker. Contrast with physical balance: on the Moon, the physical balance still correctly balances 1 kg vs 1 kg (both sides are pulled equally less) — the comparison is unchanged.
• Practical applications: bathroom scale (essentially a spring balance) shows weight (not mass). Airline luggage scales — measures weight under Earth's gravity. 'I weigh 70 kg' informally means 70 kgf (kilogram-force) = 70×9.8 N ≈ 686 N — strictly the WEIGHT, not mass (though mass is also 70 kg on Earth, the number coincides only because we define the unit). In space (ISS), a bathroom scale shows 0 even though the astronaut's mass is unchanged.

A physical (beam) balance works by comparing torques. The pan on the left has mass m₁ at distance L from the fulcrum; the pan on the right has mass m₂ at distance L. Balance condition: torque_left = torque_right → m₁ × g_eff × L = m₂ × g_eff × L → m₁ = m₂. The factor g_eff (effective gravity = g + a for upward acceleration, g − a for downward, g_moon on the Moon) CANCELS from BOTH sides. Therefore: A physical balance indicates equal masses REGARDLESS of the value of g or the acceleration of the system. Critical NEET comparison: (1) On Earth: spring balance reads W = mg; physical balance correctly compares masses. (2) On Moon: spring balance reads W/6; physical balance STILL correctly compares masses (unchanged). (3) In an accelerating elevator: spring balance reading changes; physical balance comparison is UNCHANGED.

• Practical implication: if you want to measure the TRUE MASS of an object (independent of location), use a physical balance (equal arm balance). If you want to measure the WEIGHT at a specific location, use a spring balance (calibrated in Newtons or kgf). This distinction is why scientific laboratories — for precise measurement of mass — use analytical balances (physical balances with calibrated standard masses) rather than spring balances.
• Beam balance vs equal-arm balance: a beam balance (like a traditional two-pan balance) has equal arms (L₁ = L₂ = L). If the arms are unequal (L₁ ≠ L₂), the balance condition becomes m₁L₁ = m₂L₂ (moment balance). An unequal arm balance uses known masses and the geometry to find unknown mass. NEET focuses on the equal-arm (equal L) case where g cancels cleanly.
• The 'weighing by exchange' method using unequal arm physical balance: weigh an object first on the left pan (balance with m₁), then on the right pan (balance with m₂). True mass = √(m₁×m₂) (geometric mean). This formula applies to the special case of an unequal-arm balance. For a spring balance calibrated at different orientations: if reading on one side = W₁ and inverted reading = W₂, true weight = √(W₁×W₂). This √(W₁×W₂) formula is the source of the 'W = √(W₁W₂)' NEET formula for spring balance scenarios.

A specific NEET scenario: a spring balance is clamped to a support. Weight W₁ is hung from the BOTTOM hook of the balance, and the TOP hook is attached to the fixed support (ceiling). The balance reads W₁ (equilibrium: tension in spring = W₁). Now if instead weight W₂ is hung on the TOP hook pulling DOWN (balance between two pulling masses from top and bottom): if both W₁ and W₂ pull in opposite directions through the spring — Wait, clarification: if W₁ pulls the bottom hook down and W₂ pulls the top hook down simultaneously: this is not standard configuration. Standard NEET scenario: 'a spring balance has W₁ hung from one end and W₂ from the other end' — for the balance to be in equilibrium, W₁ = W₂ = T (reading). If W₁ = W₂ = W, reading = W. If W₁ ≠ W₂, the heavier weight accelerates downward → it's NOT in equilibrium unless the support provides the difference.

• Most common NEET version: Two masses m₁ and m₂ connected by a string through a spring balance that hangs from the ceiling. The string goes over a pulley: m₁ on one side, m₂ on other, spring balance in the string. The spring balance reads the tension T = 2m₁m₂g/(m₁+m₂) (the tension in an Atwood machine). This is different from the simple single-hanging case. The spring balance reads the tension in the string, which in an Atwood machine is T = 2m₁m₂g/(m₁+m₂) — the harmonic mean type formula.
• The W = √(W₁W₂) formula appears when an UNEQUAL ARM balance is used to weigh an object: (1) place object on left pan, balance with standard mass m₁; (2) place object on right pan, balance with standard mass m₂. True mass m = √(m₁m₂). This is the 'double weighing' or 'exchange weighing' method. For spring balance with geometric mean: a spring balance reads W₁ for a body of true weight W when the spring is slightly pre-compressed (non-ideal spring), and W₂ when oriented differently; formula for true weight = √(W₁W₂). NEET typically asks this as a formula recall.
• Discrimination between instruments: NEET may ask 'which instrument will give a different reading when taken from Earth to the Moon?' Answer: Spring balance (reads W/6 on Moon). 'Which will give the same reading?' Answer: Physical (beam) balance (reads same mass). 'A body weighs 50 N on a spring balance at the equator. Taken to the poles, it weighs 50.2 N. Which is true mass instrument?' Answer: Physical balance (mass is the same at both locations; spring balance reads slightly differently due to g variation from Earth's shape and rotation).