Problem 18 - College Physics

Question

A cylindrical eraser of negligible mass is dragged across a paper at a constant velocity to the right by its pencil. The coefficient of kinetic friction between eraser and paper is \(0.650 .\) The pencil pushes down with \(4.20 \mathrm{~N}\). The height of the eraser is \(1.10 \mathrm{~cm}\) and its diameter is \(0.760 \mathrm{~cm} .\) Its top surface is displaced horizontally \(0.910 \mathrm{~mm}\) relative to the bottom. Determine the shear modulus of the eraser material.

Step-by-Step Solution

Answer

The shear modulus of the eraser material is approximately \(1.12 \times 10^6 \text{ Pa}\).

Step 1: Understanding Definitions

The shear modulus is a measure of the material's resistance to shear stress and is denoted as \( G \). It is defined as the ratio of shear stress to shear strain.

Step 2: Calculate Shear Stress

Shear stress (\( \tau \)) is calculated using the formula: \[ \tau = \frac{F}{A} \] where \( F \) is the force applied, and \( A \) is the area over which the force is applied. Here, \( F = 4.20 \text{ N} \) and \( A = \pi \times \left( \frac{0.760}{2} \right)^2 \text{ cm}^2 \).

Step 3: Compute the Area

Convert diameter from cm to m to compute the area: \[ A = \pi \times \left( \frac{0.00760}{2} \right)^2 = 4.54 \times 10^{-5} \text{ m}^2 \].

Step 4: Determine Shear Stress

Using the computed area: \[ \tau = \frac{4.20}{4.54 \times 10^{-5}} = 9.25 \times 10^4 \text{ Pa} \].

Step 5: Calculate Shear Strain

Shear strain (\( \gamma \)) is given by: \[ \gamma = \frac{x}{h} \] where \( x = 0.910 \text{ mm} = 0.000910 \text{ m} \) and \( h = 1.10 \text{ cm} = 0.0110 \text{ m} \). Thus, \[ \gamma = \frac{0.000910}{0.0110} = 0.0827 \].

Step 6: Calculate Shear Modulus

The shear modulus \( G \) is calculated using the formula: \[ G = \frac{\tau}{\gamma} \]. Substituting the values: \[ G = \frac{9.25 \times 10^4}{0.0827} \approx 1.12 \times 10^6 \text{ Pa} \].

Key Concepts

Shear Stress

Shear stress is a fundamental concept that plays a critical role in understanding how materials deform when subjected to forces. It is defined as the force applied parallel or tangential to the surface area per unit area. In essence, shear stress doesn't cause a change in volume but deforms the shape of the object.
To calculate shear stress, use the formula \[\tau = \frac{F}{A}\] where \( \tau \) is the shear stress, \( F \) is the force applied, and \( A \) is the area the force is applied over. In this exercise, the eraser is dragged across the paper with a force of \( 4.20 \text{ N} \), distributed across its circular base with a diameter of \( 0.760 \text{ cm} \).
By converting the diameter to meters, you can calculate the area \( A \) using the formula for the area of a circle: \( A = \pi \times \left( \frac{0.00760}{2} \right)^2 \), giving \( A = 4.54 \times 10^{-5} \text{ m}^2 \). From there, solving \( \tau = \frac{4.20}{4.54 \times 10^{-5}} \) leads to a shear stress of \( 9.25 \times 10^4 \text{ Pa} \).
Shear stress helps you understand how much force a material can withstand before changing its configuration.

Shear Strain

Shear strain is the measure of deformation representing the displacement between particles in the material body. It indicates how much a material will easily deform under shear stress.
Shear strain is calculated using the expression \( \gamma = \frac{x}{h} \), where:

• \( x \) is the horizontal displacement (change in position).
• \( h \) is the original height of the object being deformed.

In this situation, the eraser's top surface experiences a displacement of \( 0.910 \text{ mm} = 0.000910 \text{ m} \), while its height is \( 1.10 \text{ cm} = 0.0110 \text{ m} \).
Plugging these values into the formula gives \( \gamma = \frac{0.000910}{0.0110} = 0.0827 \). This value of shear strain expresses how the eraser's shape changes without altering its volume.
Understanding shear strain helps in discerning the flexibility or rigidity of materials when subjected to forces. It describes how much distortion occurs and assists in determining the material's mechanical properties.

Kinetic Friction

Kinetic friction refers to the resisting force that occurs when two objects move relative to each other. It is crucial in analyzing movements in both physics and engineering.
Typically, kinetic friction is calculated as \( f_k = \mu_k \times N \), where \( \mu_k \) is the coefficient of kinetic friction, and \( N \) is the normal force exerted on the object, which is equivalent to the weight in many contexts.
In the exercise provided, the pencil applies a downward force of \( 4.20 \text{ N} \) on the eraser, while the coefficient of kinetic friction between the eraser and the paper is given as \( 0.650 \). Thus, the kinetic friction can be identified as a crucial factor opposing the movement of the eraser. It contributes to the total forces that the shear stress must overcome to achieve motion.
Understanding kinetic friction helps comprehend the forces that act against motion and is a fundamental concept that enhances our insight into various mechanical systems.