Problem 63 - Physics

Question

An \(8.0-\mathrm{kg}\) stone at the end of a steel wire is being whirled in a circle at a constant tangential speed of \(12 \mathrm{~m} / \mathrm{s}\). The stone is moving on the surface of a frictionless horizontal table. The wire is \(4.0 \mathrm{~m}\) long and has a radius of \(1.0 \times 10^{-3} \mathrm{~m}\). Find the strain in the wire.

Step-by-Step Solution

Answer

The strain in the wire is approximately \(4.585 \times 10^{-4}\).

Step 1: Understand the Setup

We have a stone of mass \(8.0 \text{ kg}\) being whirled in a circle with a constant tangential speed of \(12 \text{ m/s}\). The stone is connected to a wire of length \(4.0 \text{ m}\) and radius \(1.0 \times 10^{-3} \text{ m}\). We need to determine the strain in the wire.

Step 2: Calculate the Centripetal Force

Centripetal force \(F_c\) is needed to keep the stone moving in a circle. It can be calculated using the formula \( F_c = \frac{m v^2}{r} \), where \(m = 8.0 \text{ kg}\), \(v = 12 \text{ m/s}\), and \(r = 4.0 \text{ m}\). Substituting these values, we have:\[ F_c = \frac{8.0 \times 12^2}{4.0} = \frac{8.0 \times 144}{4.0} = \frac{1152}{4.0} = 288 \text{ N} \]

Step 3: Determine the Wire's Cross-Sectional Area

The cross-sectional area \(A\) of the wire is essential to calculate the stress. It can be calculated using the formula for the area of a circle \( A = \pi r^2 \), where \(r = 1.0 \times 10^{-3} \text{ m}\). Substituting the value, we have:\[ A = \pi \times (1.0 \times 10^{-3})^2 = \pi \times 1.0 \times 10^{-6} \approx 3.14 \times 10^{-6} \text{ m}^2 \]

Step 4: Calculate the Stress on the Wire

Stress \( \sigma \) is defined as the force applied per unit area. Using the centripetal force as the applied force, stress can be calculated as:\[ \sigma = \frac{F_c}{A} = \frac{288 \text{ N}}{3.14 \times 10^{-6} \text{ m}^2} \approx 9.17 \times 10^{7} \text{ N/m}^2 \]

Step 5: Determining Young's Modulus for Steel

Young's Modulus \( E \) for steel is usually given as \( E \approx 2.0 \times 10^{11} \text{ N/m}^2 \). This value is fundamental to linking stress and strain through Hooke's Law.

Step 6: Calculate Strain in the Wire

Strain \( \epsilon \) is defined as the stress \( \sigma \) divided by Young's Modulus \( E \). Using Hooke's Law \( \epsilon = \frac{\sigma}{E} \), we find:\[ \epsilon = \frac{9.17 \times 10^7}{2.0 \times 10^{11}} \approx 4.585 \times 10^{-4} \]

Step 7: Conclusion

The calculated strain in the wire is \(4.585 \times 10^{-4}\). This dimensionless quantity indicates the proportionate elongation of the wire under the applied stress.

Key Concepts

Young's Modulus

Young's Modulus, denoted by \( E \), is an important measure of the ability of a material to resist deformation when tension or compression forces are applied. It indicates the stiffness of a solid material. In the exercise, we consider a steel wire, which is known for having a high Young's Modulus value, approximately \( 2.0 \times 10^{11} \text{ N/m}^2 \). This high value means that steel is hard to stretch or compress.

When stress is applied to a material, Young's Modulus helps determine how much it will stretch or compress. More technically, it's the ratio of tensile stress to tensile strain:

• Tensile Stress: The force applied divided by the cross-sectional area of the material.
• Tensile Strain: The change in length divided by the original length.

Knowing Young's Modulus helps predict how the material behaves under stress. It's particularly useful in engineering for choosing the right material for structures.

Strain

Strain, represented by the symbol \( \epsilon \), is a measure of deformation representing the displacement between particles in the material body relative to a reference length. In simpler terms, it indicates how much a material can stretch. It is dimensionless, meaning it has no units, and is calculated using the formula:

• \( \epsilon = \frac{\text{Change in Length}}{\text{Original Length}} \)

In the context of the problem, strain is calculated from the stress experienced by the wire due to the centripetal force acting on the stone. Once the stress \( \sigma \) is obtained, the strain can be calculated using Young's Modulus. The formula is:

• \( \epsilon = \frac{\sigma}{E} \)

This formula tells us that the strain depends directly on the stress and inversely on Young's Modulus. A higher value of Young's Modulus would result in a lower strain for the same stress, while a higher stress results in a higher strain.

Hooke's Law

Hooke's Law is a principle of physics that states that the force needed to extend or compress a spring by some distance is proportional to that distance. More formally:

• Force \( F \) is proportional to displacement \( x \). Mathematically \( F = kx \), where \( k \) is the spring constant.

Though this law primarily describes springs, it applies to many materials within the limit of elasticity — where materials return to their original shape after the force is removed.

In the context of our problem, the concept is applied using stress and strain rather than force and displacement. Hooke's Law in terms of stress and strain is expressed as:

• \( \sigma = E\epsilon \)

Here, \( \sigma \) is stress, \( E \) is Young's Modulus, and \( \epsilon \) is strain. This relationship demonstrates that within the elastic limit, stress is directly proportional to strain. Thus, knowing two of these values allows us to compute the third, which is the essence of solving problems involving elastic materials.