Shear Stress Due to Angular Deceleration

[Hobbledehoy53]

I have a rectangular block (propellar) with known width, thickness, legth, and uniform density. It is spinning at a constant angular velocity around its centroid. Then it is decelerated at a constant rate to rest. I want to find the relation between the radial distance from the centroid and the shear stress during the decelleration, so that I can determine the maximum shear stress. Please help! Thanks!

 

[Hobbledehoy53]

anyone? please...

 

[kyle8921]

Hello,

Thank you for sharing your question with me. I am happy to assist you in finding the relation between the radial distance from the centroid and the shear stress during the deceleration of a spinning rectangular block.

First, let us define some variables:

- ω (omega) = angular velocity of the block
- α (alpha) = angular deceleration of the block
- ρ (rho) = density of the block
- w = width of the block
- t = thickness of the block
- l = length of the block
- r = radial distance from the centroid of the block

To find the relation between the radial distance and the shear stress, we can use the formula for shear stress due to angular deceleration:

τ = ρω²rα

Where τ is the shear stress, ρ is the density, ω is the angular velocity, r is the radial distance, and α is the angular deceleration.

Since we know the block's width, thickness, length, and density, we can calculate the moment of inertia of the block around its centroid using the formula:

I = (1/12)ρwt³ + (1/12)ρtl³

Now, we can use Newton's second law for rotational motion to relate the angular deceleration to the moment of inertia and the torque applied to the block:

Iα = τ

Substituting the value of τ from the first equation, we get:

Iα = ρω²rα

Solving for α, we get:

α = (ρω²r)/I

Substituting this value of α in the first equation, we get the relation between the radial distance and shear stress as:

τ = (ρω²r²)/I

Now, to determine the maximum shear stress, we need to find the maximum value of r. This can be done by equating the angular momentum of the block before and after deceleration:

Iω = (1/2)Iω² + (1/2)mr²ω²

Solving for r, we get:

r = (2Iω)/(I+ml²)

Substituting this value of r in the relation we found earlier for shear stress, we get the maximum shear stress as:

τmax = (2ρω²I)/(I+ml²)

I hope this helps in finding the relation