A cylindrical steel bar is required to support a load of 650 N. the required bar must be 0.17 m in length and must not deform by more than 0.015 mm. given that the Young's modulus of steel is 210 GPa, calculate the minumum diameter of the bar.
To calculate the minimum diameter of the bar, we can use the formula for stress and strain.
1. Determine the force applied to the bar:
The force applied to the bar is given as 650 N.
2. Determine the length of the bar:
The length of the bar is given as 0.17 m.
3. Determine the deformation or change in length:
The maximum allowed deformation or change in length is given as 0.015 mm. Convert this to meters by dividing by 1000: 0.015 mm ÷ 1000 = 0.000015 m.
4. Use Hooke's Law to calculate the stress:
Hooke's Law states that stress (σ) is equal to the force (F) divided by the cross-sectional area (A) of the material: σ = F / A.
The stress (σ) can also be calculated as the Young's modulus (E) multiplied by the strain (ε): σ = E * ε.
Since we have the force (F) and the length (L) but need to find the cross-sectional area (A), we rearrange the equation to solve for A: A = F / (E * ε).
5. Plug in the values and solve for the cross-sectional area (A):
The Young's modulus of steel is given as 210 GPa. Convert this to Pascal by multiplying by 10^9: 210 GPa * 10^9 Pa/GPa = 210,000,000,000 Pa.
Plug in the values: A = 650 N / (210,000,000,000 Pa * 0.000015 m).
6. Calculate the minimum diameter:
The cross-sectional area (A) of a cylinder is given by the formula A = π * r^2, where r is the radius of the cylinder. Since we want to find the diameter, we can use the formula d = 2 * r.
We have the cross-sectional area (A) and can rearrange the formula to solve for the radius (r): r = √(A / π).
Finally, substitute the radius (r) into the formula for the diameter (d): d = 2 * √(A / π).
7. Substitute the calculated value of A into the equation and solve for the diameter:
Calculate the value of A using the above equation, and then substitute that value into the equation for the diameter.
Following these steps, you will obtain the minimum diameter of the cylindrical steel bar needed to support the load without deforming excessively.