B6. [9 Marks] 30⁰ 20140X20 DE Ofe OTO A stainless-steel orthodontic wire is applied to a tooth as shown in the diagram below. The wire has an unstretched length of 3.1 cm and a diameter of 0.22 mm. If the wire is stretched by 0.10 mm during the procedure, find the magnitude and direction of the force on the tooth. Disregard the width of the tooth and assume Young's modulus for stainless-steel is 18 × 10¹0 Nm-². ​

The magnitude of the force on the tooth is approximately 0.022 N.

To find the magnitude and direction of the force on the tooth, we can use Hooke's Law, which states that the force exerted on an object is directly proportional to the change in length of a material when it is stretched or compressed.

First, we need to calculate the strain (ε) of the stainless-steel wire.

Strain is defined as the change in length divided by the original length:

ε = ΔL / L₀

Given that the change in length (ΔL) is 0.10 mm [tex](0.10 \times 10^{-3} m)[/tex] and the unstretched length (L₀) is 3.1 cm [tex](3.1 \times 10^{-2} m)[/tex], we can calculate the strain:

[tex]\epsilon=(0.10 \times 10^{-3} m)/(3.1 \times 10^{-2} m)=0.003225[/tex]

Next, we can use Young's modulus (E) to calculate the stress (σ) in the wire.

Stress is defined as the force per unit area:

σ = E * ε

Given that Young's modulus (E) for stainless-steel is 18 × 10¹⁰ N/m², we can calculate the stress:

σ = (18 × 10¹⁰ N/m²) * 0.003225 = 5.805 × 10⁸ N/m²

Now, we can find the force (F) on the tooth by multiplying the stress by the cross-sectional area (A) of the wire:

F = σ * A

The cross-sectional area (A) can be calculated using the formula for the area of a circle:

A = π * (d/2)²

Given that the diameter (d) of the wire is 0.22 mm[tex](0.22 \times 10^{-3} m)[/tex], we can calculate the cross-sectional area:

[tex]A = \pi * (0.22 \times 10^-3 m / 2)^{2} = 3.802 \times 10^{-8} m^2[/tex]

Finally, we can calculate the force:

[tex]F = (5.805 \times 10^{8} N/m^{2}) * (3.802 \times 10^-8 m^{2}) \approx 2.206 \times 10^{-2} N[/tex]

Therefore, the magnitude of the force on the tooth is approximately 0.022 N.

Since the wire is stretched, the force is pulling the tooth in the direction opposite to the stretching.

For more questions on magnitude

https://brainly.com/question/24468862

#SPJ8

How can graphs help demonstrate the qualitative relationship that may exist in a set of data to readers?