A stainless-steel orthodontic wire is applied to a tooth, as in the figure below. The wire has an unstretched length of 3.42 cm and a radius of 0.14 mm. If the wire is stretched 0.13 mm, 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 1.8 1011 Pa.
To find the magnitude and direction of the force on the tooth, we need to calculate the amount of stress applied to the wire and then use Hooke's Law to determine the force.
Here's how we can do it step by step:
Step 1: Calculate the change in length
The change in length of the wire is given as 0.13 mm. We'll convert it to meters to match the units of the other measurements:
Change in length = 0.13 mm = 0.13 × 10^(-3) m = 1.3 × 10^(-4) m.
Step 2: Calculate the elongation
The elongation is the change in length divided by the original length:
Elongation = Change in length / Original length = 1.3 × 10^(-4) m / 3.42 × 10^(-2) m = 3.80 × 10^(-3).
Step 3: Calculate the strain
Strain is the ratio of the elongation to the original length:
Strain = Elongation / Original length = 3.80 × 10^(-3) / 3.42 × 10^(-2) = 1.11 × 10^(-1).
Step 4: Calculate the stress
Stress is the product of Young's modulus and the strain:
Stress = Young's modulus × Strain = 1.8 × 10^11 Pa × 1.11 × 10^(-1) = 1.9978 × 10^10 Pa.
Step 5: Calculate the force
Now, we can use Hooke's Law, which states that stress is directly proportional to the force applied on the wire:
Stress = Force / Area,
where Force is the force applied, and Area is the cross-sectional area of the wire.
To find the Area, we'll use the formula for the area of a circle:
Area = π × (Radius)^2.
Radius = 0.14 mm = 0.14 × 10^(-3) m = 1.4 × 10^(-4) m.
Area = π × (1.4 × 10^(-4))^2 = π × 1.96 × 10^(-8) m^2 = 6.139 × 10^(-8) m^2.
Now, rearranging the Hooke's Law equation, we can solve for Force:
Force = Stress × Area = 1.9978 × 10^10 Pa × 6.139 × 10^(-8) m^2 = 1.2267 N.
So, the magnitude of the force on the tooth is 1.2267 N. The direction of the force is along the axis of the wire.
Note: The calculation assumes linear elastic behavior and neglects the width of the tooth and any other factors that may affect the results.