Calculate the work done to twist a steel wire by an angle 45° having radius as 1 mm and length as 25 cm. Modulus of rigidity of steel is 8 × 1011 dyne/cm2.

ANSWER : To calculate the work done to twist a steel wire, we need to consider the Torsional Energy stored in the wire. The formula for the Torsional Energy (U) is given by :

U = (1/2) × (G × J) × (θ²)

where :

- U represents the torsional energy stored in the wire, measured in Joules (J).

- G is the shear modulus or modulus of rigidity of the material, measured in Pascals (Pa) or Newtons per square meter (N/m²).

- J is the polar moment of inertia of the wire cross-section, measured in square meters (m⁴).

- θ is the twist angle in radians (rad).

First, let's convert the given quantities to the appropriate units :

- Modulus of rigidity of steel (G) = 8 × 10¹¹ dyne/cm² = 8 × 10¹¹ × 10^(-5) N/m² = 8 × 10⁶ Pa.

- Radius (r) = 1 mm = 1 × 10^(-3) m.

- Length (L) = 25 cm = 25 × 10^(-2) m.

To calculate the polar moment of inertia (J) for a cylindrical wire, we use the formula :

J = (π/2) × (r⁴)

Now, substituting the values into the formula:

J = (π/2) × ((1 × 10^(-3)4)

   = π/2 × (1 × 10^(-12)) 

   = π/2 × 10^(-12) m⁴

The twist angle is given as 45°, which we need to convert to radians :

θ = 45° × (π/180)

   = π/4 radians

Now we can calculate the torsional energy (U) :

U = (1/2) × (8 × 10⁶ Pa) × (π/2 × 10^(-12) m⁴) × (π/4)²

   = (1/2) × (8 × 10⁶) × (π^2/16) × 10^(-12) J

Simplifying further, we get :

U ≈ 0.3927 J

Therefore, the work done to twist the steel wire by an angle of 45° is approximately 0.3927 Joules.

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