A piano wire has a radius of 0.50 mm.One end is fixed and the other is wrapped around a tuning peg,which has a diameter of 3.5 mm and is 80 cm away.After the wire just becomes taut,the peg is given three full turns.What is the tension in the wire? Young's modulus for steel is 2.0 × 10¹¹ N/m². A)3.8 kN B)4.3 kN C)6.5 kN D)8.7 kN E)9.8 kN

Question

Quizplus · Accepted Answer

The length of the wire can be calculated as follows:
Length of the straight wire = distance between the fixed end and the tuning peg
= 80 cm
Length of the wire wrapped around the tuning peg = circumference of the peg × number of turns
= π × diameter of the peg × number of turns
= π × 3.5 mm × 3
= 32.985 cm
Total length of the wire = length of the straight wire + length of the wrapped wire
= 80 cm + 32.985 cm
= 112.985 cm
= 1.12985 m

The cross-sectional area of the wire can be calculated as follows:
Cross-sectional area = π × (radius)²
= π × (0.50 mm)²
= 0.7854 × 10⁻⁶ m²

The tension in the wire can be calculated using the formula:
Tension = (4 × load × length) / (π × (radius)² × π × (Young's modulus))
where load is the weight applied to the wire, which in this case is the weight of the tuning peg and any additional tension required to bring the wire to pitch.

Assuming the weight of the tuning peg is negligible compared to the additional tension required to bring the wire to pitch, the load can be approximated as follows:
Load = mass per unit length × length × g
where g is the acceleration due to gravity, which is 9.81 m/s² for this problem.

The mass per unit length can be calculated using the density of steel (7.8 g/cm³) and the cross-sectional area of the wire as follows:
Mass per unit length = density × cross-sectional area
= 7.8 g/cm³ × 0.7854 × 10⁻⁶ m²
= 6.11332 × 10⁻³ kg/m

Substituting all the values into the tension formula, we get:
Tension = (4 × load × length) / (π × (radius)² × π × (Young's modulus))
= (4 × 6.11332 × 10⁻³ kg/m × 1.12985 m × 9.81 m/s²) / (π × (0.50 × 10⁻³ m)² × π × (2.0 × 10¹¹ N/m²))
= 6.472 kN
= 6.5 kN (rounded to one decimal place)

Therefore, the best choice is (C) 6.5 kN.

A Piano Wire Has a Radius of 0