QuestionCalculate the mass, in tonnes, that is supported by a cylindrical vertical pillar, 1.5 m in diameter, when the stress is . A hallow cast-iron vertical cylinder, 3 m long when unloaded, has an outer diameter of 150 mm and an inner diameter of 130 mm . Calculate (a) Maximum load that can be supported at 80 MPa and (b) the decrease in length. Assume 100GPa.
Studdy Solution
STEP 1
What is this asking?
We need to find the mass supported by a pillar under a given stress, and then figure out the max load and length change of a hollow cylinder under specific conditions.
Watch out!
Keep track of units!
We'll be jumping between millimeters, meters, Newtons, Pascals, and tonnes.
Also, don't mix up the solid pillar and the hollow cylinder.
STEP 2
1. Calculate the mass on the solid pillar.
2. Calculate the max load on the hollow cylinder.
3. Calculate the decrease in length of the hollow cylinder.
STEP 3
Alright, let's **start** by finding the area of the pillar's top.
The diameter is , so the radius is half of that, which is , or .
Remember, we need consistent units!
STEP 4
Now, the area of a circle is , so the area is .
That's a **big** area!
STEP 5
The stress is , which means for every square millimeter, there are of force.
Since we have , the total force is . **Wow**, that's a lot of Newtons!
STEP 6
To get the mass, we need to divide the force by the acceleration due to gravity, which is approximately .
So, the mass is .
STEP 7
Finally, we need the mass in tonnes.
Since , we divide by to get . **Awesome**, we found the mass supported by the pillar!
STEP 8
For the hollow cylinder, we need the cross-sectional area.
The outer radius is and the inner radius is .
STEP 9
The area is the difference between the outer circle's area and the inner circle's area: .
STEP 10
The maximum stress is , which is equal to .
The maximum load is the stress multiplied by the area: . **Excellent**, we found the max load!
STEP 11
We'll use the formula for strain: .
The stress is , and Young's Modulus is .
STEP 12
So, the strain is .
Strain is the change in length divided by the original length.
STEP 13
The original length is .
The change in length is .
But we're looking for the *decrease* in length, so it's actually or . **Perfect**, we calculated the decrease in length!
STEP 14
The mass supported by the pillar is approximately **3602.74 tonnes**.
The maximum load the hollow cylinder can support is approximately **351858.4 N**, and the decrease in length is **2.4 m**.
Was this helpful?