Solving the Cart Wheel Problem: Find Min. Tyre Diameter & Temp.

[hellboydvd]

Homework Statement

This is from Strength of Materials. A cart wheel of diameter 1.2m is to be provided with a thin steel tyre. Assuming the wheel to be rigid and if the stress in steel is not to exceed 140 MPa, find the minimum diameter of the tyre and the minimum temperature to which it should be heated before slipping it onto the wheel. Take Young's Modulus of steel=200 GPa and coeffficient of thermal expansion (alpha)= 12* 10^-6 per degree celsius

Homework Equations

alpha*(delta t)*original length=dl
Young's modulus=Stress/Strain

The Attempt at a Solution

Thought- The required diameter of the steel tyre would be less than 1.2m. This is because after heating it to a certain minimum temperature, it has to be slightly bigger than 1.2 then put on the cart wheel and cooled so that it provides a firm grip. Therefore:-

Permissible stress= 140 N/mm^2 = Strain*Young's modulus
calculate strain from this equation
strain= dl/l
dl=change in length in this case= 1.2-d ( where d is the assumed initial diameter of the steel tyre)
and l is d ( original length of the steel tyre)

Getting d from this, equate alpha*d*(delta t)= 1.2-d to get the change in temperature (Rise)

Is this approach correct or i made some error?

Thanks in advance.

 

[Jhero]

Hello,

Your approach is generally correct, but there are a few things to consider. First, it is important to note that the stress in the steel tyre should not exceed 140 MPa, not N/mm^2. This means that the units for stress should be MPa, not N/mm^2. Additionally, you should use the given value for Young's modulus, which is 200 GPa, not MPa. This will give you a more accurate calculation for the strain.

Next, you are correct in assuming that the required diameter of the steel tyre will be less than 1.2m. To find the minimum diameter, you can use the equation for strain that you have written, but it is important to remember that the strain in this case is caused by both the thermal expansion and the change in diameter. Therefore, you should add the two strains together to get the total strain, and then solve for the minimum diameter.

Once you have found the minimum diameter, you can use the equation alpha*d*(delta t) = 1.2-d to find the change in temperature. However, keep in mind that this equation assumes that the tyre is heated to the minimum temperature and then immediately placed on the wheel. In reality, the tyre may cool slightly before it is placed on the wheel, so you may need to adjust the equation to account for this.

Overall, your approach is correct, but make sure to use the correct units and to account for both thermal expansion and change in diameter when calculating the strain. Good luck with your calculations!