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The tips of nails exert tremendous pressures when hammers hit them because they exert a large force over a small area.

What force (in Newtons) must be exerted on a nail with a circular tip of 1.00 mm diameter to create a pressure of 3.00 x 10^9 Pa?

A. 0.0005
B. 9,420
C. 4,712,389
D. 2,356

Sagot :

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]
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