Comic-book superheroes are sometimes able to punch holes through steel walls.
(a) If the ultimate shear strength of steel is taken to be 3.00 x 10^9 Pa, what force is required to punch through a steel plate 1.90 cm thick? Assume the superhero's fist has cross-sectional area of 8.00 x 10^1 cm2 and is approximately circular.
To calculate the force required to punch through the steel plate, we can use the formula:
Force = (shear strength) x (area) x (thickness)
First, we need to calculate the area where the force will be applied, which is the overlapping area between the superhero's fist and the steel plate. Since the fist has a cross-sectional area of 8.00 x 10^1 cm^2 and is approximately circular, this means that the overlapping area is also approximately circular with the same area:
Area = 8.00 x 10^1 cm^2
Now we can plug this into the formula:
Force = (3.00 x 10^9 Pa) x (8.00 x 10^1 cm^2) x (1.90 cm)
But before we continue, we need to convert all measurements to the same unit. Here, we need to convert the area and thickness from cm to m:
Area = (8.00 x 10^1 cm^2) x (1 m/100 cm)^2 = 8.00 x 10^(-3) m^2
Thickness = (1.90 cm) x (1 m/100 cm) = 0.019 m
Now we plug these values back into the formula:
Force = (3.00 x 10^9 Pa) x (8.00 x 10^(-3) m^2) x (0.019 m)
Force = 456000000 N
The force required to punch through a steel plate 1.90 cm thick is approximately 456 million newtons.
To calculate the force required to punch through a steel plate, we need to apply the concept of shear stress and shear strength of the material.
Shear stress (τ) can be calculated using the equation:
τ = F / A
Where:
τ is the shear stress,
F is the force applied perpendicular to the surface, and
A is the cross-sectional area of the object.
In this case, the superhero's fist is approximately circular, so we can calculate the cross-sectional area (A) using the formula for the area of a circle:
A = π * r^2
Where:
A is the cross-sectional area, and
r is the radius of the circular cross-section (half of the diameter of the fist).
Now that we have the cross-sectional area, we can calculate the force (F) required to punch through the steel plate using the equation for shear stress:
F = τ * A
Given data:
Shear strength of steel (τ) = 3.00 x 10^9 Pa
Steel plate thickness (d) = 1.90 cm
Fist cross-sectional area (A) = 8.00 x 10^1 cm^2
Let's calculate the force required:
Step 1: Convert the units to SI units (meters).
Steel plate thickness (d) = 1.90 cm = 0.019 m
Fist cross-sectional area (A) = 8.00 x 10^1 cm^2 = 8.00 x 10^3 m^2
Step 2: Calculate the radius of the fist.
Since the fist is approximately circular and we have the cross-sectional area, we can use the formula for the area of a circle.
A = π * r^2
Rearranging the formula:
r^2 = A / π
r = √(A / π)
Substituting the values:
r = √(8.00 x 10^3 m^2 / π)
Calculate the radius using a calculator.
Step 3: Calculate the force required.
Using the equation for shear stress:
F = τ * A
Given:
Shear strength of steel (τ) = 3.00 x 10^9 Pa
Calculate the force:
F = (3.00 x 10^9 Pa) * (8.00 x 10^3 m^2)
Once again, use a calculator to get the result.
The resulting force will give you the answer to part (a) of the question.