A drive shaft of a belt conveyor system transmits 12kW of power at a speed of 40 rpm. If the shaft diameter is 50mm * 120mm long, calculate a)The shear stress in the shaft.

b)The angle of twist if modulus of rigidity =80 GN/m2
c)If the shear stress is limited to 60 Mn/m2 what must the shaft diameter be increased to.
Any help with this question would be very helpful

To solve this problem, we can use the formulas for shear stress, angle of twist, and the relation between shear stress and shaft diameter. Let's go through each part step by step.

a) Shear stress in the shaft:
Shear stress (τ) can be calculated using the formula:
τ = (16 * Power * 1000) / (π * ω * d^3)

Where:
Power = 12 kW = 12,000 W (converted to watts)
ω = Angular velocity = (40 rpm * 2π rad/1 min) / 60 = 4.18879 rad/s (converted to radians per second)
d = diameter of the shaft in meters (given in mm, so we need to convert it to meters)

Let's substitute the given values into the formula:
τ = (16 * 12,000) / (π * 4.18879 * (0.05/2)^3)

Calculating this will give us the shear stress in the shaft.

b) Angle of twist:
The angle of twist (θ) can be calculated using the formula:
θ = (L * Power * 1000) / (π * G * d^4)

Where:
L = Length of the shaft = 120 mm = 0.12 m (converted to meters)
Power = 12,000 W
G = Modulus of rigidity = 80 GN/m^2 = 80 × 10^9 N/m^2
d = diameter of the shaft in meters

Substituting the given values into the formula:
θ = (0.12 * 12,000) / (π * 80 × 10^9 * (0.05/2)^4)

Calculating this will give us the angle of twist.

c) Increasing the shaft diameter:
In order to determine the required increase in the shaft diameter to limit the shear stress to 60 MN/m^2 (60 × 10^6 N/m^2), we use the formula for shear stress:
τ = (16 * Power * 1000) / (π * ω * d^3)

We can rearrange this formula to solve for d:
d = ((16 * Power * 1000) / (π * ω * τ))^1/3

Substituting the given values into the formula:
d = ((16 * 12,000) / (π * 4.18879 * 60 × 10^6))^1/3

Calculating this will give us the increased shaft diameter required to limit the shear stress.

By following these steps and performing the necessary calculations, you can find the answers to each part of the question.