Answer :
The angle of twist for the shaft is approximately 0.027 radians. To find the angle of twist for the shaft, we can use the formula: θ = (TL) / (GJ)
Where θ is the angle of twist, T is the torque being transmitted, L is the length of the shaft, G is the shear modulus, and J is the polar moment of inertia.
The polar moment of inertia for a solid circular shaft can be calculated using the formula:
J = π/2 ×(d^4)
Where d is the diameter of the shaft.
Plugging in the values given in the question, we have:
T =?
L = 1.5m
G = 80GPa = 80 x 10^9 Pa = 80 x 10^9 N/m^2
d = 200mm = 0.2m
First, we need to convert the shear stress into torque.
The maximum shear stress can be found using the formula:
τ_max = T_max / (J_max × 0.5 r)
Where J_max is the maximum polar moment of inertia, and r is the radius of the shaft.
Since the diameter is given, we can find the radius:
r = d/2 = 0.1m
Plugging in the values, we can solve for T_max:
318 x 10^6 Pa = T_max / ((π/2 × (0.2^4)) × 0.5 × 0.1)
Simplifying the equation, we find that T_max = 4.032 x 10^7 Nm
Finally, we can substitute the values into the formula for θ:
θ = (4.032 x 10^7 Nm × 1.5m) / (80 x 10^9 N/m^2 × π/2 × (0.2^4))
Calculating the right side of the equation, we find that θ ≈ 0.027 radians
Therefore, the angle of twist for the shaft as it transmits the torque is approximately 0.027 radians.
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