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Maharashtra State Board > Science > Physics Question Bank > Elasticity > You are Here |

### Very Short Answers (1 Mark)

**Q1. What is plasticity? Write an example of perfect plasticity.**

- Plasticity is the property of a material to undergo a permanent deformation even after removal of external deforming forces.
- Wall putty can be considered as a perfectly plastic body.

**Q2**. **What are the factors, which decide ductility and brittleness of the material?**

- Materials which have a great plastic range get stretched to long thin wires before they break are called ductile materials. Hence thin wires can be formed using ductile materials. e.g. steel, aluminum, gold, copper, silver etc.
- Few materials break quite suddenly as soon as the stress-strain curve starts deviating from the straight line after the elastic limit. They are called brittle materials. Hence thin wires cannot be formed using brittle materials. e.g. glass, ceramics, cast iron etc.

**Q3. Explain the terms in elasticity.**

**Point of Proportionality:**The point of proportionality is the point on the stress-strain curve above which the stress in a material is no longer linearly proportional to strain i.e. it is the point above which Hooke’s law is not obeyed. Point of proportionality is also called proportional limit.**Elastic Limit:**It is the maximum stress to which an elastic body can be subjected without causing permanent deformation is called the elastic limit.**Yield Point:**The point on the stress-strain curve at which the strain begins to increase without any increase in the stress is called the yield point.**Breaking Stress:**The maximum stress which can be applied to a wire is called breaking stress.**Perfectly Elastic Bodies:**Those bodies which regain their original shape, size or both completely after removal of the deforming force are called perfectly elastic bodies.**Perfectly Plastic Bodies:**Those bodies which retain their new shape, size or both completely after removal of the deforming force are called perfectly plastic bodies.

**Q4. State Hooke’s law of elasticity and hence define modulus of elasticity.**

**Statement of Hooke’s Law:**Within the elastic limit, the stress is directly proportional to the strain.**Modulus of Elasticity:**The ratio of the stress produced in a body to corresponding stress produced in it is called the modulus of elasticity of the material of the body.

**Q5. State and define modulus of elasticity applicable to the substance in all states of matter.**

- The bulk modulus of elasticity is applicable to the substance in all states of matter.
- Within the elastic limit, the ratio of volumetric stress to the corresponding volumetric strain in a body is always constant, which is called as Bulk modulus of elasticity.

**Q6. What are linear elastic or Hookean materials?**

- The materials which obey Hooke’s law are called elastic or Hookerian materials.

**Q7. Draw a stress against strain curve for an elastic body.**

**Q8. How is the flexibility of a rope increased?**

- In order to increase the flexibility of a rope, it is made up of a large number of thin wires, braided together.

**Q9. Which materials are called elastomers?**

- The substances such as tissue of aorta rubber etc. which can be stretched to cause large strains are called elastomers.

**Q10. Define stress and strain.**

**Stress:**Stress is defined as an internal elastic restoring force per unit cross-sectional area of a body. OR stress is defined as applied force per unit cross-sectional area of a body.**Strain:**The change in dimension per unit original dimension of a body subjected to deforming forces is called as Strain.

**Q11. Define modulus of elasticity.**

- Within the elastic limit, the ratio of volumetric stress to the corresponding volumetric strain in a body is always constant, which is called as Bulk modulus of elasticity.

**Q12. Stress and pressure have the same dimensions but pressure is not the same as stress. Why?**

- The term pressure is often used with fluids (gases or liquids), whereas term stress is more often used with solids.
- The pressure only acts perpendicular to a surface, whereas stress can also be parallel (shear stress) to a surface as well as perpendicular to it.

**Q13. Define compressibility and state its SI unit.**

- The reciprocal of bulk modulus of elasticity is called as compressibility.
- Its S.I. unit is m
^{2}N^{-1}or Pa^{-1}

**Q14. In the method for determination of Young’s modulus of the material of wire, what is the use of reference wire?**

- Since a long wire is used, a small change in temperature during the course of the experiment will produce a measurable change in length of the wire due to thermal expansion, then the measured extension will be greater than the actual extension in the wire.
- This error is eliminated by using a reference (dummy) wire. As both experimental wire and reference wire are of the same material and same original length, the change in length due to change in temperature will be the same for both the wires and thus there will be no shift in the position of the bubble in spirit level.

**Q15. Which modulii of elasticity are related to a jelly cube?**

- As jelly cube has volume, bulk modulus is related with it similarly it can undergo a change in shape, hence modulus of rigidity is also associated with it.

**Q16. What are the requirements of ultimate stress while designing a rope?**

- The ropes are designed such that the ultimate stress does not exceed the breaking stress and elastic limit.

**Q17. Explain the terms ductility and malleability.**

**Ductility:**Ductility is a mechanical property of a material by which material (metal) can drawn into wires.**Malleability:**Malleability is a mechanical property of a material by which material (metal) can be hammered or pressed into thin sheets.

**Q18. Why Young’s modulus and shear modulus are relevant only for solids and not for fluids?**

- Definition of both Young’s modulus and modulus of rigidity involves the change in size and shape (geometry) of the body. Now only solids possess definite geometry shape while fluids do not have definite geometric shape. Hence Young’s modulus and shear modulus are relevant only for solids and not for fluids

### Short Answers I (2 Marks)

**Q1. Derive an expression for modulus of rigidity.**

- Consider a rigid body as shown in the figure which is fixed along the surface CD. Let it be acted upon by tangential force F along surface AB as shown. Let lateral surface AD get deflected through angle θ as shown. The tangential force F per unit area of surface AB is called as shear stress.

**Q2. Derive an expression for modulus of elasticity related to change in length.**

- The modulus of elasticity related to the change in length is Youn’s modulus of elasticity.

- Consider a wire of length ‘L’ and radius of cross-section ‘r’ is fixed at one end and stretched by suspending a load of ‘mg’ from the other end. Let ‘‘ be the extension produced in the wire when it is fully stretched.

- Now, by the definition of Yong’s modulus of elasticity we have This is an expression for Young’s modulus of elasticity of a material of a wire.

This is an expression for Young’s modulus of elasticity of a material of a wire.

**Q3. Derive an expression for compressibility of fluid.**

Volumetric Stress = Load / Area = Pressure Intensity = dP

Volumetric strain = – Change in volume (dV)/ Original Volume (V)

The negative sign indicates the decrease in the volume

- Within the elastic limit, the ratio of volumetric stress to the corresponding volumetric strain in a body is always constant, which is called as Bulk modulus of elasticity.

- The reciprocal of bulk modulus of elasticity is called as compressibility. considering the magnitude only

This is an expression for compressibility of fluid.

**Q4 and 5.** Distinguish between Young’s modulus, bulk modulus, and modulus of rigidity.

Young’s Modulus of Elasticity |
Bulk Modulus of Elasticity |
Modulus of Rigidity |

Within the elastic limit, it is the ratio of longitudinal stress to longitudinal strain | Within the elastic limit, it is the ratio of volumetric stress to volumetric strain | Within the elastic limit, it is the ratio of shear stress to shear strain |

It is associated with the change in the length of a body. | It is associated with the change in the volume of a body. | It is associated with the change in the shape of a body. |

It exists in solid material bodies | It exists in solids, liquids, and gases. | It exists in solids only. |

It is a measure of the stiffness of a solid material | It determines how much the body will compress under a given amount of external pressure. | It describes an object’s tendency to shear |

Young’s modulus of the material of a wire is given by |
The bulk modulus of the material of a body is given by | The shear modulus of the material of a body is given by |

**Q6. Distinguish between plasticity and elasticity.**

Elasticity |
Plasticity |

The body exhibiting elasticity regains its shape or size after removal of the external force. | The body exhibiting plasticity retains its new shape or size after removal of the external force. |

There is a temporary change in dimensions of the body on application of the deforming force. | There is a permanent change in dimensions of the body on application of the deforming force. |

Internal restoring force is set up inside the body. | No Internal restoring force is set up inside the body. Or they are very negligible. |

The ratio of stress to corresponding strain produced is constant. | The ratio of stress to corresponding strain produced is not constant. |

materials exhibiting elasticity: Steel, rubber, etc | Materials exhibiting plasticity: PVC, plaster of Paris, wax, etc. |

**Q7. Explain the origin of elasticity in solids**

- When a body is stretched the inter-atomic spacing increases and when it is compressed the inter-atomic spacing decreases.

- In both the cases, internal forces are created in the body which tend to restore the atoms back to their original positions. Such internal forces are called internal elastic forces or restoring forces.
- The magnitude of restoring or internal elastic force is the same as applied force.
- These restoring forces are responsible for the elastic property of the body.

**Q8. The block in the given diagram rests on the ground. Which face A, B or C experiences a) the largest stress b) the smallest stress when the block is resting on it?**

- In all the cases the force (F) is equal to the weight of the block. It is constant. Now, the stress is inversely proportional to the area on which the force is acting.
- The area of face A is 30 x 10 = 300 cm
^{2},the area of face B is 20 x 10 = 200 cm^{2}and the area of face C is 30 x 20 = 600 cm^{2}. - As the area of face B is the smallest one the stress is maximum when the block is lying on face B.
- As the area of face C is the largest one the stress is minimum when the block is lying on face C.

**Q9. Two cylinders shown in diagram are identical in all respects except one is hollow. When identical forces are applied to the right end of each cylinder, explain which cylinder streches the most.**

- In both the cases, the force (F) is the same. Now, the stress is inversely proportional to the area on which the force is acting.
- In case of the hollow cylinder, the area of crosssection is less. Hence stress developed in it will be more. Thus strain produced in it will be more. Hence the hollow cylinder stretches the most.

**Q10. Discuss the factors on which the bending of a beam having a rectangular cross-section, depends on when the beam is loaded at the centre.**

The sag or extent of bending of a beam is given by

- Thus the sag (δ) of the beam is
- directly proportional to the load (W)
- directly proportional to the cube of length (l) of the beam.
- inversely proportional to the breadth (b) of the beam
- inversely proportional to the cube of the depth (d) of the beam
- inversely proportional to Young’s modulus (Y) of elasticity of the beam.

**Q11. A metallic rod is heated. Show that the thermal stress is directly proportional to its coefficient of linear expansion and Young’s modulus of the material of the rod.**

- If a bar which is heated and prevented from expansion or heated rod is prevented from contraction as it cools, then stress is produced in the bar. This stress is called a thermal stress.
- If bar is prevented from expansion or contraction, the change in length is given by

Δl = l ∝ Δθ

Where, Δl = change in length prevented.

l = original length of the rod

Δθ = Change in temperature.

∝ = coefficient of linear expansion of the material of the rod.

Where Y = Young’s modulus of elasticity.

This is an expression for the thermal stress.

We can observe that the thermal stress is directly proportional to

its coefficient of linear expansion and Young’s modulus of the material of the rod.

**Q12. Define Poisson’s ratio. What are the limits of Poisson’s ratio for a practical isotropic material?**

- The ratio of transverse strain to the corresponding longitudinal strain is called Poisson’s ratio.
- For homogeneous isotropic medium -1 ≤ m ≤ 0.5
- Poisson’s ratio of cork is zero, that of metal is 0.3 and that of rubber is 0.5

**Q13. Draw the stress-strain curve for elastic tissue of Aorta and discuss the conclusions about elastic properties of the aorta.**

- The graph shows a stress-strain curve for the elastic tissue of aorta, present in the heart. The elastic region is very large. But it is not a straight line. It means the material of aorta does not obey Hooke’s law over most of the region. Similarly, there is not a well-defined plastic region. Substances like tissue of aorta can be stretched to cause large strains are called elastomers

**Q14. What are the advantages of Ι ****beam?**

- The I beam minimizes both weight and stress of the beam. An I beam is much lighter and is strong for bending.
- I beam has more load bearing surface, which prevents buckling and is enough to reduce too much buckling.
- Thus I beam provides high bending moment and lot of material is saved.

**15. Why hollow circular pole or tube are preferred over solid circular poles?**

- A hollow circular pole or tube is resistant to bending in all the directions than the solid circular pole of the some mass, but smaller radius, since it has greater effective cross-section.

### S.A.II (3 Marks)

**Q1. What is strain energy? Derive an expression for strain energy per unit volume when the wire is loaded.**

- The work done by external applied force during stretching is stored as potential energy (U) in the wire and is called as strain energy in the wire.
- Consider a wire of length ‘L’ and area of cross-section ‘A’ be fixed at one end and stretched by suspending a load ‘M’ from the other end. The extension in the wire takes place so slowly that it can be treated as quasi-static change; because internal elastic force in the wire is balanced by the external applied force and hence acceleration is zero.
- Let at some instant during stretching the internal elastic force be ‘f’ and the extension produced be ‘x’. Then,

- Since at any instant, the external applied force is equal and opposite to the internal elastic force, we can say that the work done by the external applied force in producing a further infinitesimal dx is

- Let ‘ l ‘ be the total extension produced in the wire, and work done during the total extension can be found by integrating above equation.

This is an expression for the work done in stretching wire.

- The work done by external applied force during stretching is stored as potential energy (U) in the wire and is called as strain energy in the wire. Thus the strain energy is given by

- This is an expression for strain energy or potential energy per unit volume of stretched wire.

**Q2. Derive an expression for work done when the wire is loaded. What is strain energy?**

- Consider a wire of length ‘L’ and area of cross-section ‘A’ be fixed at one end and stretched by suspending a load ‘M’ from the other end. The extension in the wire takes place so slowly that it can be treated as quasi-static change; because internal elastic force in the wire is balanced by the external applied force and hence acceleration is zero.
- Let at some instant during stretching the internal elastic force be ‘f’ and the extension produced be ‘x’. Then,

- Since at any instant, the external applied force is equal and opposite to the internal elastic force, we can say that the work done by the external applied force in producing a further infinitesimal dx is

- Let ‘ l ‘ be the total extension produced in the wire, and work done during the total extension can be found by integrating above equation.

This is an expression for the work done in stretching wire.

- The work done by external applied force during stretching is stored as potential energy (U) in the wire and is called as strain energy in the wire.

**Q3. Draw a stress against strain graph for a ductile material under increasing load and hence explain the behaviour of wire.**

- The behaviour of wire under increasing load can be studied using Searle’s apparatus. The wire whose behaviour is to be studied is used in the apparatus, at the free end, increasing loads are applied. For each load, stress and strain are calculated. Then the behaviour of wire is studied by plotting a graph, stress versus strain.

- For ductile material, the graph is as shown. From O to A graph is a straight line which clearly indicates that the stress is directly proportional to strain, which indicates that Hooke’s Law is obeyed in this region. Point A is called as the limit of proportionality.
- The elastic limit is the point up to which the Hooke’s law is applicable. Stress corresponding to this is called the elastic limit. If the load is removed before the elastic limit is crossed, then the wire will be able to recover its original length completely.
- If the load is further increased, we get curve AA’ which indicate that Hooke’s law is not obeyed. The extension starts increasing faster than the load, and the graph bends towards strain axis. If the wire is strained up to a point A’ and then if the load is removed, the wire is not able to recover its original length. However, the wire still retains its elastic properties. We can see it by the fact, that when the load is steadily reduced, a new straight line graph such as A’O’ is obtained. In this case, the wire undergoes permanent deformation. The corresponding permanent strain OO’ is called permanent set or permanent strain or residual strain.
- If the load is increased further, a point B is reached, at which the tangent to the curve becomes parallel to strain axis. It indicates that there is the extension in the wire without an increase in the load. Here wire exhibit plastic flow. The point B is called as yield point and corresponding stress is called as yield stress.
- Initially, as wire elongates area of cross-section decreases uniformly, but if the wire is loaded beyond point B, stress at some local point starts increasing rapidly due to neck formation in that region and ultimately wire breaks. This point is called as the breaking point, and corresponding stress is called as breaking stress or ultimate stress or ultimate strength.
- For ductile material, there is neck formation at breaking point C. Before breaking ductile material always show plastic flow. For obtaining an appreciable extension of wire in Serle’s experiment, the specimen wire should be long and thin.

**Q4. Within elastic limit, prove that Young’s modulus of the material of wire is the stress required to double its length.**

As the length is doubled, change in length = *l* = 2L – L = L

Strain = *l* / L = L/L = 1

Now, Young’s modulus of elasticity is given by

Y = Stress / Strain = Stress/1 = Stress

- Hence within elastic limit, Young’s modulus of the material of wire is the stress required to double the length of the wire.

**Q5. Derive an expression for strain energy per unit volume and show that, the strain energy per unit volume is proportional to Young’s modulus of the material of the wire.**

- Consider a wire of length ‘L’ and area of cross-section ‘A’ be fixed at one end and stretched by suspending a load ‘M’ from the other end. The extension in the wire takes place so slowly that it can be treated as quasi-static change; because internal elastic force in the wire is balanced by the external applied force and hence acceleration is zero.
- Let at some instant during stretching the internal elastic force be ‘f’ and the extension produced be ‘x’. Then,

- Since at any instant, the external applied force is equal and opposite to the internal elastic force, we can say that the work done by the external applied force in producing a further infinitesimal dx is

- Let ‘ l ‘ be the total extension produced in the wire, and work done during the total extension can be found by integrating above equation.

This is an expression for the work done in stretching wire.

- The work done by external applied force during stretching is stored as potential energy (U) in the wire and is called as strain energy in the wire. Thus the strain energy is given by

- This is an expression for strain energy or potential energy per unit volume of stretched wire.

Now. Young’s modulus of elasticity for a material of a wire is constant.

Thus, strain energy per unit volume ∝ (stress)^{2}

i.e. strain energy per unit volume is directly proportional to the square of the stress.

- Thus the strain energy per unit volume is proportional to Young’s modulus of the material of the wire.

**Q6. Derive the relation between Young’s modulus, thermal stress, and coefficient of linear expansion.**

- If a bar which is heated and prevented from expansion or heated rod is prevented from contraction as it cools, then stress is produced in the bar. This stress is called a thermal stress.
- If bar is prevented from expansion or contraction, the change in length is given by

Δl = l ∝ Δθ

Where, Δl = change in length prevented.

l = original length of the rod

Δθ = Change in temperature.

∝ = coefficient of linear expansion of the material of the rod.

Where Y = Young’s modulus of elasticity.

This is the required relation.

**Q7. Describe an experiment to determine Young’s modulus of the material of thin wire.**

- Two identical wires A and B are suspended from a rigid support so that the points of suspension are very close to each other. Searle’s apparatus blocks are attached to the lower ends of the wires by means of chucks F
_{1}and F_{2}. - Searle’s apparatus block consists of two metal frames P and Q. The two frames are loosely connected by cross strips in such a way that the frame Q can move relatively with respect to frame P. A spirit level S is hinged to the frame P and is rested on the tip of a micrometer screw M which can work in a nut fixed in the frame Q.
- At the lower end, each frame carries a hanger from which slotted weights can be suspended. Wire A is dummy wire from which a fixed load of about 1 kg (dead weight) is suspended.

**Procedure:**

**Initial readings and settings:**- Initially, the length (L) of wire B is measured. Its mean radius (r) is found with the help of micrometer screw gauge. The wire A is experimental wire, it is initially subjected to a sufficient load called ‘zero load’ (about 1 kg) to avoid kinks in the wire. Micrometer screw is adjusted to bring the bubble in the spirit level at the centre and the reading is noted. This is called ‘zero reading’.
**Loading the wire :**- The load suspended from wire B is then increased in equal steps of about 0.5 kg-wt. let ‘m’ be the mass in the hanger. Each time, after waiting for about two minutes, the bubble is brought to the centre by rotating the screw and micrometer reading is noted. This is extension or elongation (l
_{1}) in the wire. This way five to six readings are taken. **Unloading the wire :**- After loading procedure is complete the wire is unloaded in the same steps of 0.5 kg-wt and the readings ( l
_{2}) are noted again for each step.

**Calculations:**

- The mean of the readings for loading ( l
_{1}) and unloading ( l_{2}) is calculated and represented as (l) for each step. Then Young’s modulus of the material is calculated in each step using formula,

- The average value of Young’s Modulus (Y) is calculated. A care should be taken to avoid possible errors.

**Precautions:**

- Following are sources of errors and care should be taken to avoid or minimize them.
- Error due to kinks in the wire.
- Errors due to a backlash of the screw.
- Error due to bending (yielding) of the support.
- Error due to thermal expansion or contraction
- Error due to the crossing of the elastic limit and/or slipping of the wire from the chucks.

**Q8. Answer the following questions with reference to the graph for material A and B.**

**a) Which material has greater Young’s modulus?**

**b) Which material is more ductile?**

**c) Which material is more strong?**

- From graph Young’s modulus of material = (Stress/Strain) = slope of the straight portion of the graph. The straight portion of the graph of material A is steeper than the straight portion of the graph of material B. Hence slope of the straight portion of the graph of material has more slope than that for material B. Hence Young’s modulus for material A is greater than it is for material B.
- A large part of the horizontal portion of the graph of material A is parallel to the strain axis. Thus Material A is more ductile tan material B.
- The strength of the material is determined by the amount of stress required to cause a fracture. Fracture point is the extreme point in a stress-strain curve. The extreme point of material A is well above the extreme point of material B hence it can beconcluded that material A can withstand more strain than material B. Hence, material A is stronger than material B.

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