Problem 14 - College Physics

Question

A stainless-steel orthodontic wire is applied to a tooth, as in Figure P9.14. The wire has an unstretched length of \(3.1 \mathrm{~cm}\) and a radius of \(0.11 \mathrm{~mm}\). If the wire is stretched \(0.10 \mathrm{~mm}\), 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 \(18 \times 10^{10} \mathrm{~Pa}\).

Step-by-Step Solution

Answer

The magnitude of the force on the tooth is \(22 N\) and it acts in the direction opposite to the stretching of the wire.

Step 1: Calculate the cross-sectional area

First, we need to calculate the cross-sectional area of the wire using the formula for the area of a circle \(A = \pi r^2\). The radius is given as 0.11 mm, but we need to convert this to meters. So, \( r = 0.11 mm = 0.11 \times 10^{-3} m\). Therefore, \( A = \pi (0.11 \times 10^{-3})^2 = 3.8 \times 10^{-8} m^2\).

Step 2: Calculate the stretch in meters

Next, we need to convert the stretch of 0.10 mm to meters: \( \Delta L = 0.10 mm = 0.10 \times 10^{-3} m\).

Step 3: Substitute into the Hooke's law formula

The formula derived from Hooke's law is \( F = \frac{Y \cdot A \cdot \Delta L}{L}\). Substituting the values into the formula, we get \( F = \frac{(18 \times 10^{10} Pa) \cdot (3.8 \times 10^{-8} m^2) \cdot (0.10 \times 10^{-3} m)}{3.1 \times 10^{-2} m}\).

Step 4: Simplify the expression

Simplify the expression to find the magnitude of the force. Thus, \( F = 22 N \).

Step 5: Determine the direction of the force

The direction of the force will be in the same direction as the stretch. This is because materials resist deformation, and thus the force exerted by the wire will oppose the stretch. Hence, the direction of the force is in the opposite direction of the stretching.

Key Concepts

Young's Modulus

Young's Modulus is a fundamental concept in material science. It's a measure of the stiffness of a solid material. When we talk about stiffness, we mean how much a material resists deformation due to an applied force. Young's Modulus is defined as the ratio of stress (force per unit area) to strain (proportional deformation in length) in the range where they are proportional.

This property is essential when we want to understand how materials like metals behave when they are under stress. For example, in the case of the stainless steel orthodontic wire, Young's Modulus tells us how the steel will respond when a stretching force is applied.

• Young's Modulus formula: \( Y = \frac{F \cdot L}{A \cdot \Delta L} \)
• It is measured in Pascals (Pa).
• The higher the Young's Modulus, the stiffer the material.

A material with a high Young’s Modulus does not stretch or compress easily, meaning it is very rigid. Whereas, a lower Young’s Modulus indicates a material is more elastic and can deform easily when stressed.

Stress and Strain

Stress and Strain are two related concepts that describe what happens when forces are applied to a material.

**Stress** is defined as the force exerted per unit area within materials. It is the internal resistance offered by a body to an external force, which tends to deform it. Imagine pulling on a piece of rubber – the tension in the rubber due to your pull is stress.

• Formula for Stress: \( \sigma = \frac{F}{A} \)
• Measured in Pascals (Pa) or Newtons per square meter (N/m²).

**Strain**, on the other hand, is a measure of deformation representing the displacement between particles in the material body. In simple terms, it measures how much a material deforms in response to stress.

• Formula for Strain: \( \varepsilon = \frac{\Delta L}{L} \)
• It is a dimensionless quantity as it is a ratio of lengths.

It's important to understand that while stress considers the force's magnitude, strain looks at how the object stretches or compresses as a result of stress. Together, they help us understand the behavior of materials when they undergo extension, compression, or bending.

Elasticity in Materials

Elasticity is the property that allows materials to return to their original shape after being deformed by an external force. When you stretch or compress something and it goes back to its original form, that's elasticity at work.

Materials that have high elasticity can undergo significant deformation without permanent change. Common examples of elastic materials include rubber bands and certain metals like steel, which return to their initial form when forces are removed.

• Elastic Limit: Beyond this point, permanent deformation occurs.
• Hooke's Law: Describes how the force needed to extend or compress a spring by some distance is proportional to that distance. Formula: \( F = k \cdot x \)

In the case of orthodontic wires, the elasticity is crucial. It ensures that the wire applies a controlled amount of force on the teeth, helping them move to the desired position over time. Elasticity in engineering materials ensures that they can withstand loads within their elastic limits without breaking.