Respuesta :
Answer:
The length of the wire is approximately 3.4593 meters
Explanation:
The given parameters of the torsion pendulum are;
The radius of the steel holding the sphere, r = 0.25 mm = 0.00025 m
The diameter of the sphere, d = 3 cm = 0.03 m
The modulus of rigidity of steel, N = 80 GPa
The density of the material of the sphere, ρ = 11,300 kg/m³
The period of oscillation = 2 seconds
The volume of the sphere, V = 4/3 × π × R³
∴ V = 4/3 × π × (0.03/2)³ = 0.00001413716 m³
The mass of the sphere, m = V × ρ = 0.00001413716 × 11,300 ≈ 0.15975 kg
The mass of the sphere, m ≈ 0.15975 kg
The moment of inertia of a sphere, I = 2/5×m×R²
∴ I = 2/5×m×R² = 2/5 × 0.15975 kg × (0.03/2 m)² = 1.43775 × 10⁻⁵ kg·m²
The relationship between the rigidity modulus of the steel, 'N', and the length of the wire, 'l', is given as follows;
[tex]N = \dfrac{8 \times\pi \times I \times l}{T^2 \times r^4}[/tex]
Where;
l = The length of the wire
Therefore, we have;
[tex]l =\dfrac{ N \times T^2 \times r^4 }{8 \times\pi \times I }[/tex]
By plugging in the values, we get;
[tex]l =\dfrac{ 80 \times 10^9 \times 2^2 \times 0.00025^4 }{8 \times\pi \times 1.43775 \times 10^{-5} } \approx 3.4593[/tex]
The length of the wire, l ≈ 3.4593 meters.
