Maximum Stress in a rotating Cylinder.

[mk47k]

Homework Statement

A cylinder has a height and length of 0.89m. It's mass is known to be 1000kg. The cylinder is rotated at 14000 RPM. Calculate the Maximum stress in the cylinder.

2. The attempt at a solution
The maximum stress should occur at the center of the cylinder.
The centepietal force is equal to F=mv² / r .

Now this is where I'm having problems. After I calculate the force what area should I divide it by to find the max stress?. Can I just assume the area to be really small such as 0.001*d ?

 

[taureau20]

what's d answer?

 

[Mene]

I would approach this problem by first defining the variables and assumptions. The maximum stress in a rotating cylinder can be calculated using the equation σ = ρω^2r, where ρ is the density of the material, ω is the angular velocity, and r is the distance from the axis of rotation. In this case, we can assume that the cylinder is made of a uniform material, so ρ can be considered a constant.

Next, we need to determine the angular velocity (ω) and the distance from the axis of rotation (r). The given information states that the cylinder is rotated at 14000 RPM, which can be converted to radians per second (rad/s) by multiplying by 2π/60. This gives us an angular velocity of approximately 1468.6 rad/s.

To find the distance from the axis of rotation (r), we can use the given dimensions of the cylinder. Since the cylinder is rotating on its central axis, we can assume that the maximum stress occurs at the center of the cylinder. Therefore, the distance from the axis of rotation (r) is equal to half of the height or length of the cylinder, which is 0.445 m.

Now, we can plug in the values into the equation σ = ρω^2r to calculate the maximum stress. Using the given mass of 1000 kg, we can find the density of the cylinder by dividing the mass by its volume. Since the cylinder has a height and length of 0.89 m, the volume can be calculated as V = πr^2h, where r is the radius of the cylinder. Since the cylinder is not given to have a specific radius, we can assume it to be half of the height, which is 0.445 m. This gives us a volume of approximately 0.558 m^3. Therefore, the density (ρ) is equal to 1000 kg/0.558 m^3 = 1792.85 kg/m^3.

Plugging in all the values, we get σ = (1792.85 kg/m^3)(1468.6 rad/s)^2(0.445 m) = 1.46 x 10^9 Pa. This is the maximum stress in the rotating cylinder.

In conclusion, as a scientist, I would approach this problem by defining the variables and assumptions, and then using the appropriate equation to calculate