The length of a light string is 1.4 m when the tension on it is 5 N . If the tension increases to 7 N , the length of the string is 1.56 m . The original length of the string is__________m.

Solution

Physics sir · 4 steps

Extension, not length, grows with tension, so two readings give two equations in the natural length.

1. Set up Hooke's law. Let the original length be L₀. Within the elastic limit, the extension is proportional to tension, so L = L₀ + kT, where k is the extension per newton.
2. Write both readings. At 5 N: 1.4 = L₀ + 5k. At 7 N: 1.56 = L₀ + 7k.
3. Find k. Subtract the first equation from the second: 2k = 0.16 m, so k = 0.08 m N⁻¹.

   L₀ = 1.4 - 5 × 0.08 = 1.4 - 0.4 = 1 m

4. So. The original length of the string is 1 m. As a check, 1 + 7 × 0.08 = 1.56 m, which matches the second reading.

The common slip: 0.6. Taking the extension per newton as 0.16 m (the change for 2 N) instead of 0.08 m gives 1.4 - 0.8 = 0.6, not 1.

The idea to keep. For tensions T₁, T₂ giving lengths L₁, L₂, the natural length is L₀ = (T₂ L₁ - T₁ L₂)/(T₂ - T₁). Remember, extension, not total length, is proportional to tension.

Answer from the NTA final answer key. Solution by PrepWiser.