To solve this problem, we need to calculate the force required to create a specific pressure given the diameter of the nail's circular tip. Here's a step-by-step solution:
1. Convert Diameter from Millimeters to Meters:
The diameter of the circular tip given is 1.00 mm. To work with standard SI units, we need to convert this to meters. Remember that 1 mm = 0.001 meters.
[tex]\[
\text{Diameter} = 1.00 \text{ mm} = 1.00 \times 0.001 \text{ m} = 0.001 \text{ m}
\][/tex]
2. Calculate the Radius:
The radius is half of the diameter.
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{0.001 \text{ m}}{2} = 0.0005 \text{ m}
\][/tex]
3. Calculate the Area of the Circular Tip:
The area [tex]\( A \)[/tex] of a circle is given by the formula [tex]\( A = \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius.
[tex]\[
A = \pi \times (0.0005 \text{ m})^2
\][/tex]
Simplifying the expression:
[tex]\[
A \approx 3.14159 \times 0.00000025 \text{ m}^2
\][/tex]
[tex]\[
A \approx 7.854 \times 10^{-7} \text{ m}^2
\][/tex]
4. Calculate the Force:
Force ([tex]\( F \)[/tex]) can be calculated using the relationship between pressure ([tex]\( P \)[/tex]), force, and area. The formula is [tex]\( P = \frac{F}{A} \)[/tex]. Rearranging this to solve for force:
[tex]\[
F = P \times A
\][/tex]
Given the pressure [tex]\( P = 3.00 \times 10^9 \)[/tex] Pa:
[tex]\[
F = 3.00 \times 10^9 \text{ Pa} \times 7.854 \times 10^{-7} \text{ m}^2
\][/tex]
[tex]\[
F \approx 2356.194490192345 \text{ N}
\][/tex]
Therefore, the force that must be exerted on the nail with a circular tip of 1.00 mm diameter to create a pressure of [tex]\( 3.00 \times 10^9 \)[/tex] Pa is approximately [tex]\( 2,356 \)[/tex] Newtons. So, the correct answer is:
[tex]\[
\boxed{2,356}
\][/tex]
