Practice Questions

Propose a method to distinguish between a brittle material and a ductile material by analyzing their respective stress-strain curves. Create sketches of representative curves to support your proposal.

Justify the inclusion of a negative sign in the formula for Bulk Modulus, $B = -\frac{p}{\Delta V / V}$.

Compare the compressibility of solids, liquids, and gases. Explain the reason for the vast difference in their compressibility based on their molecular structure.

Examine why strain is a dimensionless quantity.

Define the term 'stress' and name its SI unit.

Recall the formula for Young's modulus ($Y$) in terms of force ($F$), area ($A$), original length ($L$), and change in length ($\Delta L$).

Justify why Bulk modulus is relevant for solids, liquids, and gases, whereas Young's modulus and Shear modulus are relevant only for solids.

Name the three types of elastic moduli.

List the three types of strain and briefly describe each.

Define Bulk Modulus and explain the significance of the negative sign in its formula.

Define elastomers and give an example.

A copper wire of length $4.0$ m stretches by $2.0$ mm. Calculate the longitudinal strain.

A steel wire of length $2.0$ m and cross-sectional area $2.5 \times 10^{-6} \text{ m}^2$ is stretched by a force of $500$ N. Calculate the elongation of the wire. The Young's modulus for steel is $2.0 \times 10^{11} \text{ N m}^{-2}$.

Examine why Hooke's law is not universally applicable to all materials, giving an example of a material that does not obey it.

Analyze the stress-strain curves for two materials, Material A and Material B. Material A has a longer plastic region and a higher ultimate tensile strength than Material B. Which material is more ductile and which is stronger? Justify your answer.

Calculate the pressure required to compress a solid block of steel by $0.1\%$. The bulk modulus of steel is $160 \times 10^9 \text{ N m}^{-2}$.

Summarize the key differences between elasticity and plasticity.

Identify the property that determines whether a material is ductile or brittle on a stress-strain graph.

Critique the classical mechanics concept of a 'rigid body' using the principles of elasticity discussed in this chapter. Justify why this concept is an idealization.

Formulate a relationship to compare the compression of a steel rod and an aluminum rod, both having the same length and cross-sectional area, when they are subjected to the same compressive force. Justify which rod will compress more.

Justify the engineering practice of using I-shaped girders for constructing bridges instead of solid rectangular girders of the same mass and material.

Explain Hooke's law and define the modulus of elasticity.

Describe hydraulic stress and the type of strain it produces.

A steel rod is stretched by a force of $500$ N. If the cross-sectional area of the rod is $2.5 \times 10^{-6} \text{ m}^2$, calculate the tensile stress on the rod.

Compare and contrast Young's modulus, Shear modulus, and Bulk modulus in terms of the type of stress and strain they relate to.

Apply the concept of elasticity to explain why steel is considered more elastic than rubber.

Calculate the elastic potential energy stored per unit volume in a steel wire that is stretched to a strain of $1.0 \times 10^{-3}$. The Young's modulus of steel is $2.0 \times 10^{11} \text{ N m}^{-2}$.

Analyze the region beyond the elastic limit on a stress-strain curve. What is meant by 'permanent set' and how is it represented on the graph?

Evaluate the common statement: 'Rubber is more elastic than steel.' Justify your conclusion using the scientific definition of elasticity and the concept of Young's modulus.

A student claims that the work done in stretching a wire is simply the final stretching force multiplied by the total elongation. Critique this approach and formulate the correct expression for the elastic potential energy stored per unit volume.

Evaluate the stress-strain curves for two materials, A and B. Material A has a higher ultimate tensile strength but fractures soon after reaching it. Material B has a lower ultimate tensile strength but exhibits a much larger plastic region. Propose which material is better suited for making car bodies and justify your choice.

A solid cube is subjected to a shearing force $F$ on its top face, while its bottom face is riveted to the floor. Formulate the expression that relates the angular displacement ($\theta$) of the side faces to the applied force ($F$), the area of the face ($A$), the length of the cube's side ($L$), and its shear modulus ($G$).

Create a problem involving a composite rod made of a steel wire of length $2.0 \text{ m}$ and a copper wire of length $1.5 \text{ m}$ joined end-to-end. Both wires have a diameter of $2.0 \text{ mm}$. If a mass of $10 \text{ kg}$ is suspended from the free end of the copper wire, formulate the step-by-step calculation to find the total elongation of the composite rod. Use $Y_s = 2.0 \times 10^{11} \text{ N/m}^2$ and $Y_c = 1.1 \times 10^{11} \text{ N/m}^2$.

Summarize the concept of elastic potential energy in a stretched wire and recall the formula for elastic potential energy per unit volume.

Design an experiment to determine the Young's modulus of a material in the form of a long wire. List the apparatus required, the procedure to be followed, the observations to be recorded, and formulate the final calculation.

A deep-sea submersible is designed to operate at a depth where the pressure is $1000$ atm. Its viewing port is a solid glass disc with a Bulk Modulus of $B = 37 \text{ GPa}$. Evaluate the fractional change in volume ($\Delta V/V$) of the glass port at this depth and critique whether this compression is a significant concern for the structural integrity. (Use $1 \text{ atm} = 1.013 \times 10^5 \text{ Pa}$).

Solve for the vertical deflection of the top face of a copper cube with an edge length of $20$ cm. One face is fixed to a horizontal surface, and a tangential force of $5.0 \times 10^4$ N is applied to the opposite face. The shear modulus of copper is $42 \times 10^9 \text{ N m}^{-2}$.

Two wires, one of steel and one of brass, each $1.5$ m long and with a diameter of $2.0$ mm, are connected end to end. When a load of $10$ kg is suspended from the free end, solve for the total elongation of the combined wire. (Given: $Y_{\text{steel}} = 2.0 \times 10^{11} \text{ N m}^{-2}$, $Y_{\text{brass}} = 0.91 \times 10^{11} \text{ N m}^{-2}$, and $g = 9.8 \text{ m/s}^2$)

Propose a design for a structural beam intended to support a heavy, moving load, such as in a bridge. Justify your choice of material and cross-sectional shape based on the principles of elastic behavior.

A solid sphere of radius $10$ cm is placed deep in the ocean where the pressure is $2.0 \times 10^7$ Pa. Solve for the change in the volume of the sphere. The bulk modulus of the material of the sphere is $1.0 \times 10^{11} \text{ N m}^{-2}$.

Explain the terms 'yield point' and 'ultimate tensile strength' with reference to a stress-strain curve.

A crane is designed to lift a maximum load of $20$ metric tons ($20,000$ kg). The steel cable used has a yield strength of $400 \times 10^6 \text{ N m}^{-2}$. To ensure safety, the maximum stress should not exceed $25\%$ of the yield strength. Calculate the minimum diameter required for the cable. (Take $g = 10 \text{ m/s}^2$).

You are tasked to design a steel cable for a crane to safely lift a maximum load of $20,000$ kg. The yield strength of the steel to be used is $\sigma_y = 250 \times 10^6 \text{ N/m}^2$. Propose a suitable radius for the cable, incorporating a safety factor of 10, and justify your design choice.

Explain why steel is considered more elastic than rubber, even though rubber stretches more easily.

Analyze why the beams used in the construction of bridges and buildings often have an I-shaped cross-section. How does this shape help in preventing bending and buckling while being cost-effective? Use the formula $\delta = W l^3 / (4 b d^3 Y)$ to support your analysis.