A die is designed to punch holes of radii 2.17 x 10-2 m in a metal sheet that is 5.53 x 10-3 m thick. To punch through the sheet, the die must exert a shearing stress of 3.5 x 108 Pa. What force must be applied to the die? (Hint: Consider carefully which area is used in your calculation.)
Well, isn't that just a hole lot of fun? Let's get calculating!
First, we need to determine the area that the force needs to be applied to. Since we're punching holes, we'll be focusing on the area of the circle. The formula for the area of a circle is A = πr^2, where r is the radius.
In this case, the radius of the hole is given as 2.17 x 10^-2 m. So let's plug in the numbers and calculate the area:
A = π(2.17 x 10^-2)^2
Now, we multiply this area by the thickness of the metal sheet, which is given as 5.53 x 10^-3 m. This will give us the volume of the material that needs to be punched through:
Volume = A x thickness
Now, we need to find the force required. The force is equal to the stress multiplied by the area:
Force = stress x area
But since we already calculated the volume, we can modify the equation:
Force = stress x volume / thickness
Now, let's insert the given stress value of 3.5 x 10^8 Pa, and the values we calculated earlier:
Force = (3.5 x 10^8 Pa) x [(π(2.17 x 10^-2)^2) x (5.53 x 10^-3 m)] / (5.53 x 10^-3 m)
After performing the calculations, you should get your answer for the force. So go crunch those numbers and let me know what you come up with!
To determine the force required to punch through the metal sheet, we can use the formula for shearing stress:
Shearing stress = Force / Area
In this case, the shearing stress is given as 3.5 x 10^8 Pa and we need to find the force.
First, let's determine the area that should be used in the calculation. Since the die is designed to punch holes of radius 2.17 x 10^-2 m, the area of the hole can be calculated as:
Area = π * (radius)^2
Area = π * (2.17 x 10^-2)^2
Now, let's calculate the area of the hole by substituting the values:
Area = 3.14 * (2.17 x 10^-2)^2
Next, we need to consider the thickness of the metal sheet. The force required will be applied over the entire thickness of the sheet. Therefore, the effective area for the calculation should be:
Effective Area = Area of the hole * Thickness of the sheet
Effective Area = [(3.14 * (2.17 x 10^-2)^2) * (5.53 x 10^-3)]
Now, let's calculate the effective area:
Effective Area = 3.14 * (2.17 x 10^-2)^2 * 5.53 x 10^-3
Now, we can rearrange the formula for shearing stress to solve for force:
Force = Shearing stress * Effective Area
Substituting the given values:
Force = 3.5 x 10^8 Pa * (3.14 * (2.17 x 10^-2)^2 * 5.53 x 10^-3)
Now, let's calculate the force:
Force = 3.5 x 10^8 Pa * (3.14 * (2.17 x 10^-2)^2 * 5.53 x 10^-3)
To find the force required to punch through the metal sheet, we need to use the formula for shearing stress:
Shearing stress = Force / Area
First, we need to calculate the area of the hole in the metal sheet. The area of a circle is given by the formula:
Area = π * r^2
Given that the radius of the hole is 2.17 x 10^-2 m, we can calculate the area:
Area = π * (2.17 x 10^-2)^2
Next, we need to consider which area to use in the calculation. Since the sheet is being punched through, we should use the area of the hole. This will give us the maximum shearing stress required.
Now, we can rearrange the formula to solve for force:
Force = Shearing stress * Area
Given that the shearing stress is 3.5 x 10^8 Pa, and we calculated the area of the hole, we can substitute these values into the equation:
Force = (3.5 x 10^8) * (π * (2.17 x 10^-2)^2)
Now we can calculate the force by plugging in the values and solving the equation.