Comic-book superheroes are sometimes able to punch holes through steel walls.

(a) If the ultimate shear strength of steel is taken to be 2.90 109 Pa, what force is required to punch through a steel plate 2.10 cm thick? Assume the superhero's fist has cross-sectional area of 1.05 102 cm2 and is approximately circular.
N

To determine the force required to punch through a steel plate, we can use the formula for shear stress:

Shear stress = Force / Area

Given:
Thickness of the steel plate (d) = 2.10 cm
Ultimate shear strength of steel (τ) = 2.90 * 10^9 Pa
Cross-sectional area of the fist (A) = 1.05 * 10^2 cm^2

First, let's convert the thickness from centimeters to meters:
d = 2.10 cm = 2.10 * 10^-2 m

Next, let's convert the area from square centimeters to square meters:
A = 1.05 * 10^2 cm^2 = 1.05 * 10^-4 m^2

Now, rearrange the formula for shear stress to solve for force:
Force = Shear stress * Area

Substituting the given values into the equation:
Force = (2.90 * 10^9 Pa) * (1.05 * 10^-4 m^2)

Performing the calculation:
Force = 3.045 * 10^5 N

Therefore, the force required to punch through a steel plate 2.10 cm thick is approximately 3.045 * 10^5 N.