Deformation of Solids: Physics 9702 (Cambridge International AS & A Level)

A spring hangs vertically from a fixed support. When a load of $6.0\text{ N}$ is hung from the free end, the spring stretches by $4.0\text{ cm}$. The spring obeys Hooke's law throughout.

What is the spring constant of the spring?

Two springs, P and Q, each obey Hooke's law over the range of forces considered. Spring P has spring constant $200\text{ N m}^{-1}$ and spring Q has spring constant $300\text{ N m}^{-1}$.

(a) The springs are joined end-to-end, so that the same force acts through both springs and the total extension is the sum of the individual extensions (a series combination). Determine the effective spring constant of this series combination. [2]

(b) The same two springs are instead arranged side-by-side between two rigid plates, so that both springs are forced to have the same extension when a load is applied (a parallel combination). Calculate the effective spring constant of this parallel combination. [2]

(c) A load of $12\text{ N}$ is hung from the parallel combination described in (b). Calculate the extension produced. [2]

A technician investigates a metal wire of original length $2.50\text{ m}$ and uniform diameter $0.80\text{ mm}$. One end of the wire is clamped and the wire hangs vertically. A load of $45\text{ N}$ is hung from the free end, producing an extension of $1.5\text{ mm}$. The wire does not exceed its limit of proportionality.

(a) Calculate the cross-sectional area of the wire. [2]

(b) Calculate the stress in the wire when the $45\text{ N}$ load is applied. [2]

(c) Calculate the strain in the wire when the $45\text{ N}$ load is applied. [2]

(d) Determine the Young modulus of the material of the wire. [2]

A student fixes one end of a thin metal wire and hangs increasing loads from the free end, recording the extension produced at each load. A graph of load $F$ (vertical axis) against extension $x$ (horizontal axis) is plotted from the results, with the following features.

• From the origin O, the graph is a straight line up to a point labelled A, where $F_A = 8.0\text{ N}$ and $x_A = 2.0\text{ mm}$.
• Beyond A, the graph curves very slightly, but a short sample loaded to any point up to a further point labelled B (only slightly beyond A) is found to return exactly to its original length once the load is removed.
• The wire is then loaded further, well beyond B, to a point labelled C, where $F_C = 12.0\text{ N}$ and $x_C = 4.0\text{ mm}$. The load is then removed completely. Once fully unloaded, the wire is measured again and is found to be $1.0\text{ mm}$ longer than its original, unstretched length.

(a) State the name given to point A, and state the feature of the graph between O and A that identifies it. Calculate the spring constant of the wire for the region between O and A. [2]

(b) State what is meant by the elastic limit. A short sample of the wire is stretched only as far as a point between A and B, and the load is then removed. State and explain whether this sample returns to its original length, and hence state the name given to point B. [3]

(c) Calculate the elastic potential energy stored in the wire when it has been loaded, within the region obeying Hooke's law, up to point A. [2]

(d) State the term used to describe the type of deformation that has occurred in the wire between B and C, given that a permanent extension of $1.0\text{ mm}$ remains after the load is fully removed. Calculate the extension of the wire, measured at C (before unloading), that was recovered elastically once the load was removed. [2]

A spring obeys Hooke's law and has a spring constant of $40\text{ N m}^{-1}$. The spring is first stretched from its natural length to an extension of $5.0\text{ cm}$, and is then stretched further to an extension of $9.0\text{ cm}$. The spring remains within the region where it obeys Hooke's law throughout.

What is the additional elastic potential energy stored in the spring as the extension increases from $5.0\text{ cm}$ to $9.0\text{ cm}$?

A spring obeys Hooke's law and has a spring constant of $25\text{ N m}^{-1}$. The spring is stretched from its natural length by an extension of $12\text{ cm}$.

What is the force applied to the spring?

A student is asked to determine the Young modulus of the material of a metal wire.

(a) Describe how the student could carry out an experiment, using a long thin wire and standard laboratory apparatus, to obtain the measurements needed to determine the Young modulus. Your answer should include how the load, extension, original length and cross-sectional area are each obtained. [4]

(b) State one precaution the student should take when measuring the diameter of the wire, and explain why this precaution improves the accuracy of the final value obtained for the Young modulus. [2]

(c) In one trial, the wire has original length $1.80\text{ m}$ and diameter $0.36\text{ mm}$. A load of $18\text{ N}$, applied within the limit of proportionality, produces an extension of $0.90\text{ mm}$. Calculate the Young modulus of the wire from this data. [3]

A metal wire is stretched by a steadily increasing force until it has been deformed well beyond its elastic limit. The force is then removed completely.

Which statement correctly describes what happens to the wire once the force has been removed?

A steel support strut in a model bridge has an original (unloaded) length of $0.800\text{ m}$ and a uniform circular cross-section of diameter $6.0\text{ mm}$. When a compressive force of $2.8\text{ kN}$ is applied along its length, the strut shortens by $0.40\text{ mm}$. The strut obeys Hooke's law throughout this compression, with stress and strain related in the same way as for extension.

(a) Calculate the cross-sectional area of the strut. [2]

(b) Calculate the compressive stress in the strut when the $2.8\text{ kN}$ force is applied. [2]

(c) Calculate the compressive strain in the strut when the $2.8\text{ kN}$ force is applied. [2]

(d) Determine the Young modulus of the steel. [2]

(e) Calculate the elastic strain energy stored in the strut at this compression. [2]

A student investigates a spring by hanging different loads from it and measuring the total (stretched) length of the spring for each load. The unstretched (natural) length of the spring is $12.0\text{ cm}$. The results are shown in the table.

(a) Calculate the extension of the spring for each load, and use your values to determine the spring constant of the spring while it obeys Hooke's law. [3]

(b) State, with a reason based on your values from (a), the load at which the spring's behaviour first becomes inconsistent with Hooke's law. [2]

(c) Calculate the elastic potential energy stored in the spring when the load is $3.0\text{ N}$. [2]