Download Physics mechanics Notes and more Study notes Computer science in PDF only on Docsity!
Elasticity
Elasticity: The property of a body due to which it can regain its original configuration after the removal of deforming force is called elasticity. For example, a stretched rubber band returns to its original state when the force is removed. Perfectly Elastic Body: A body which regains its original configuration completely and instantaneously after the removal of deforming force is called perfectly elastic body. There is no perfectly elastic body in nature. So it is an ideal concept only. A quartz fibre is nearest approach to the perfectly elastic body. Plasticity: The property of a body due to which it cannot regain its original configuration after the removal of deforming force is called plasticity. For example, wet clay exhibits plasticity. Perfectly plastic Body: A body which does not regain its original configuration at all after the removal of deforming force is called perfectly plastic body. There is no perfectly plastic body in nature. So it is also an ideal concept only. For example paraffin wax, wet clay, putty etc. are nearest approach to the perfectly plastic body. Stress:
When a deforming force is applied to a body, an internal force called restoring force is developed within the body which opposes the deformation. The restoring force developed per unit area of the body is called stress. The restoring force is equal and opposite to the deforming force. So stress can also be defined as the deforming force per unit area of the body. i.e.
Stress = 𝐷𝑒𝑓𝑜𝑟𝑚𝑖𝑛𝑔 𝑓𝑜𝑟𝑐𝑒 𝑜𝑟 𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝑓𝑜𝑟𝑐𝑒 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦 SI unit of stress is N/m^2 and CGS unit is Dyne/ cm^2
Types of Stress:
Depending upon how force is applied on a body stress is generally divided into following three different types.
1. Normal stress:
If the deforming force is applied perpendicularly to the surface of a body, then the stress is called normal stress. The normal stress is called tensile stress or compression stress according as the stress causes increase in length or decrease in length.
2. Tangential stress or shearing stress:
If the deforming force is applied parallel to the surface of a body, then the stress is called tangential stress or shearing stress.
F
3. Bulk stress or volume stress or hydraulic stress:
When a material is subjected to an external pressure that changes its volume uniformly without changing shape, then the stress is called bulk stress or volume stress. For example when a body is immersed in a liquid, pressure gives the bulk stress.
Difference between Pressure and stress:
Pressure is always normal to the area while stress can be normal or tangential. Pressure is always compressive while stress can be compressive as well as tensile. Pressure is a scalar while stress is a tensor.
Strain:
The ratio of change in configuration to the original configuration under the application of deforming force is called strain. i.e.
Strain = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑐𝑜𝑛𝑓𝑖𝑔𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑐𝑜𝑛𝑓𝑖𝑔𝑢𝑟𝑎𝑡𝑖𝑜𝑛 It is a unit less quantity.
Types of strain:
Depending upon the types of change in configuration strain is divided into following three different types.
1. Longitudinal strain:
The ratio of change in length (∆𝑙) to the original length (l ) under the application of normal stress is called longitudinal strain. i.e.
Longitudinal strain = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑙𝑒𝑛𝑔𝑡ℎ 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ =^
∆𝑙 𝑙 =^
𝑒 𝑙
2. Volumetric strain:
The ratio of change in volume ( ∆v ) to the original length (v) under the application of normal stress is called longitudinal strain. i.e.
Volumetric strain = Change in volume Original volume =^
∆v v
V
F
F
F
F
removal of the deforming force. The point E in the curve up to which the body regains its original configuration after the removal of deforming force is called elastic limit. The region OPE represents the elastic behaviour of the material of the wire. iii. If the wire is stretched beyond elastic limit E, (i.e., in the region EA) the strain increases rapidly without significant increase in stress. The point A is called the yield point and the corresponding stress is called the yield stress. After E (up to A) the wire will not recover its original configuration. At this stage, after unloading, it will not retrace the curve to O. Rather it returns to 𝑂′. along A𝑂′. There is a permanent extension (deformation) in the wire called permanent set given by O𝑂′. The yield point A is sometimes called commercial elastic limit. iv. Beyond A (in the region AB), a very small increase in stress produces a very large increase in strain. At B, the stress is maximum. The maximum stress corresponding to B is called breaking stress. After reaching point B, the strain increases even if the stress is decreased. In the region BC the wire literally flows and finally breaks at C called breaking point. The region EABC represents the plastic behaviour of the material of the wire.
Note: If the plastic region between E to B is large, the material is said to be ductile and can be drawn into wires. If it is small, it is called brittle as it will break soon after the elastic limit is crossed.
Elastomers: The materials for which stress strain variation is not a straight line within the elastic limit and strain produced is comparatively much larger than the stress applied are called elastomers. Such materials have no plastic range and breaking point lies very close to elastic limit. For example; rubber is an elastomer.
Hooke’s law: It states that the restoring force developed on a wire is directly proportional to the extension or compression produced in it if the proportional limit is not exceeded. i.e.
F ∝ e
Where, F is restoring force and e is extension or compression produced.
Alternatively Hooke’s law can be stated as follows.
“Within the proportional limit, the stress developed in an elastic body is directly proportional to the strain produced in it.”
i.e., Stress ∝ Strain
or, Stress = E × 𝑆train, where E is constant called modulus of elasticity.
∴ E = 𝑆𝑡𝑟𝑒𝑠𝑠 𝑆𝑡𝑟𝑎𝑖𝑛 Thus, the modulus of elasticity can be defined as the ratio of stress and strain within the proportional limit. Its SI unit is N/m^2.
Types of moduli of elasticity:
Depending upon the types of stress applied and resulting strain, there are three types of modulus of elasticity which are as follows.
i. Young’s modulus of elasticity:
Within the proportional limit, the ratio of normal stress to the longitudinal strain is called Young’s modulus of elasticity. It is denoted by Y.
i.e., Y = 𝑁𝑜𝑟𝑚𝑎𝑙 𝑆𝑡𝑟𝑒𝑠𝑠 𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛
or, Y = 𝐹/𝐴 𝑒/𝑙
or, Y = 𝐹𝑙 𝐴𝑒 …………….(i)
If e = l, then Y = F A = Stress. Hence, Young’s modulus of elasticity of a material is numerically equal to the stress which will double its length.
ii. Bulk modulus of elasticity:
Within the proportional limit, the ratio of normal stress to the volume strain is called bulk modulus of elasticity. It is denoted by B or K.
i.e., B = 𝑁𝑜𝑟𝑚𝑎𝑙 𝑆𝑡𝑟𝑒𝑠𝑠 𝑆𝑡𝑟𝑎𝑖𝑛
or, B = ∆𝑃 (−∆𝑉𝑉 )
, where ∆𝑃 is normal stress or increase in pressure
or, B = - V ∆𝑃 ∆𝑉 …………….(ii) Compressibility: The reciprocal of bulk modulus of elasticity is called compressibility. It is denoted by C.
i. e., C = 1 𝐵 = -^
1 𝑉 (
∆𝑉 ∆𝑃)
iii. Shear modulus or modulus of rigidity:
Within the proportional limit, the ratio of tangential stress to the shear strain is called shear modulus or modulus of rigidity. It is denoted by 𝜂.
i.e., 𝜂 = 𝑇𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙 𝑆𝑡𝑟𝑒𝑠𝑠 𝑆ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛
or, 𝜂 = 𝐹/𝐴 𝜃
or, 𝜂 = 𝐹 𝐴𝜃 ……………..(iii)
Poisson’s ratio:
When a wire is stretched by applying a force, its length increases and diameter decreases. The strain produced along the direction of force is called longitudinal strain and strain produced along the direction perpendicular to the applied force is called lateral strain.
Consider a wire of length l and diameter d is stretched by applying a force so that its length increases by e and diameter decreases by ∆𝑑.
Then, longitudinal strain (𝛼) =
𝑒 𝑙 (length increases)
And lateral strain (𝛽) = - Δ𝑑 𝑑 , (diameter decreases)
Experimentally it has been found that within the elastic limit, lateral strain is directly proportional to the longitudinal strain.
i.e. 𝛽 ∝ 𝛼
or, 𝛽 = 𝜎𝛼, where 𝜎 is constant called Poisson’s ratio.
∴ 𝜎 = 𝛽 𝛼
i.e., 𝜎 = - ∆d/d 𝑒/𝑙
The theoretical value of Poisson’s ratio lies between -1 and 0.5 while practical value lies in between 0 and 0.5. Poisson’s ratio gives the decrease in lateral size per unit increase in length of the wire within the elastic limit.
Experimental verification of Hooke’s law (Determination of Young’s modulus of elasticity):
O
Slope = 𝑏 𝑎
Extension (e)
b a
Load (F)
P Q
M V
Dead load
Standard weights
Fig : Experimental arrangement for the verification of Hook’s law.
The experimental arrangement consists of two wires a reference wire P and an experimental wire Q suspended vertically on a rigid support. The reference wire P carries the main scale and experimental wire Q caries vernier scale of the vernier apparatus at the lower ends as shown in figure. A small load called dead load is suspended at the lower end of the reference wire to make it free from the kinks. A scale pan attached at the lower end of the experimental wire also makes the experimental wire free from the kinks. Before starting the experiment, the initial reading (main scale and vernier scale) of the vernier apparatus is noted. Now without exceeding the elastic limit, weights are gradually added on the scale pan in steps and corresponding readings are noted. The extension produced for the different values of load applied on the experimental wire are calculated. A plot of load applied versus extension produced will be a straight line passing through origin which verifies Hooke’s law.
To determine the Young’s modulus of elasticity, the slope of the graph is calculated. Now Young’s modulus of elasticity of the material of the wire is given by
Y =
𝐹𝑙 𝐴𝑒
or, Y = 𝐹 𝑒 ×^
𝑙 𝐴
or, Y = 𝑆𝑙𝑜𝑝𝑒 × 𝑙 𝐴
or, Y = 𝑆𝑙𝑜𝑝𝑒 × 𝑙 𝜋𝑟^2
Length l and radius r of the experimental wire can be measured with the help of meter scale and screw gauge. Then Young’s modulus of elasticity of the material of the wire can be calculated.
Elastic after effect: We know that the elastic bodies regain their original configuration after the removal of deforming force. But some bodies regain their original configuration immediately after the removal of deforming force and some bodies take time to regain their original configuration. The time delay in regaining the original configuration by the elastic body after the removal of a deforming force is called elastic after effect. It is least for quartz or phosphor bronze and maximum for glass. Due to this region, suspensions made from quartz or phosphor bronze are used in moving coil galvanometer.
Elastic fatigue: A temporary loss of elastic properties due to continuous use of a body for long time is called elastic fatigue. For example, when a copper wire is bent once, it may not break but breaks when bent repeatedly at the same location. It is due to elastic fatigue developed in the copper wire at the given location.
or, Y =
𝑇 ⁄𝐴 Δ𝑙 ⁄𝑙
𝑇 ⁄𝐴 𝛼 ∆𝜃 or, 𝑇 𝐴 =^ 𝑌𝛼^ ∆𝜃^ ………………(ii) This is the expression for thermal stress. The tension developed in the material due to restriction in thermal expansion or contraction is given by T = YAα ∆θ ……………….(ii)
Factors affecting elasticity:
1. Effect of temperature: When temperature of a body is raised, the distance between its atoms increases. As a result of it, the elastic restoring force decreases which in turn decreases the elasticity of the body. It means the modulus of elasticity decreases with increase in temperature. a) In case of invar (alloy of nickel and steel) the value of modulus of elasticity remains practically unchanged by the change of temperature. b) The lead becomes elastic like steel when cooled in liquid air. c) A carbon filament which is highly elastic at ordinary temperature becomes plastic when heated by passing current through it. d) In case of a rubber, the modulus of elasticity (Y) increases with increase in temperature. 2. Addition of impurities: The addition of impurity may increase or decrease the modulus of elasticity of the body. If the impurity to be added in a metal is more elastic than the metal, the elasticity of the resulting metal increases. If the impurity to be added is more plastic than the metal, then the elasticity will be reduced. 3. Annealing: It means slow cooling after heating. It reduces the elastic property of the body. 4. Hammering or rolling : It increases the elasticity of the body. Conceptual questions: 1. Steel is more elastic than rubber, why?
Ans: A body is said to be more elastic if it has larger value of modulus of elasticity. When a steel wire and a thread of rubber having same length and same area of cross section are stretched by applying same force, the extension produced in the rubber (er) is greater than the extension produced in the steel (es). The Young’s modulus of elasticity of steel and rubber can be expressed as
Ys = 𝐹𝑙 𝐴𝑒𝑠^ ……………….(i) And Yr = 𝐹𝑙 𝐴𝑒𝑟^ ……………….(i) Dividing (i) by (ii) we get, F
F
𝑌𝑠 𝑌𝑟^ =^
𝑒𝑟 𝑒𝑠^ ……………(iii)
Since er > es, Ys > Yr
i.e., Young’s modulus of elasticity of steel is greater than that of rubber and hence it is more elastic
2. Which modulus determines the stretching of coiled spring? 3. Ans: The modulus of elasticity that determines the stretching of coiled spring is the shear modulus or modulus of rigidity. 4. Is it possible to double the length of metal wire by applying a force over it? Ans: No, it is not possible. Because within elastic limit the strain is only of the order of 10 −3. Wires actually break much before it is stretched to double the length. 5. Why bridges are declared unsafe after long use? Ans : Due to repeated loading and unloading for long period of time, the bridge gets elastic fatigue. i.e., the elastic property of the bridge is lost. After the bridge gets elastic fatigue, larger strain is produced even with smaller loads and ultimately the bridge may collapse. So, bridges are declared unsafe after long use. 6. Why are steel cables made by twisting many thin wires together? Ans : When a metal is repeatedly deformed, its resistance to plastic deformation increases. This phenomenon is called work hardening. In order to draw metals into wire, large plastic deformation is required. This process hardens the wire. As a result the thin wires have higher yield stress than the rod. Therefore the mechanical strength of of steel cable made by twisting many thin wires together is greater than the cable of steel rod of the same diameter 7. Explain in terms of breaking stress why elephant has thicker legs than that of man? 8. Why are rubbers used as vibration absorber? 9. Why are springs made of steel not of copper? 10. Water is more elastic than air, why? 11. Why do spring balance show wrong reading after long use? 12. The stress strain graphs for two materials A and B are shown in figure. The graphs are drawn to the same scale. a) Which material has greater Young’s modulus? b) Which material is more ductile? c) Which is more brittle? d) Which of the two is strong material?
A B
Stress
Strain Strain
Stress
16. A 1.5 m long steel rod is fixed at both ends and its temperature is increased by 50°C. If the coefficient of linear expansion of steel is 1.2× 10 −5^ 𝐾−1^ and its Young’s modulus is 2.0×10^11 N/m2., determine the thermal stress developed in the rod. (Ans: 120 MPa) 17. Calculate the greatest length of a wire than can hang vertically without breaking. (Given: Breaking stress of a material = 108 N/m^2 and density of material 3000 kg/m^3 ) (Ans : 3.4 km ) 18. A material has a Poisson’s ratio 0.2. If a uniform rod of it suffers longitudinal strain 4 × 10 −3, deduce the percentage change in its volume. [Hint: Fractional change in volume, ∆𝑉 𝑉 = (1 - 2𝜎)^
∆𝑙 𝑙 ]^ (Ans : 0.24 %)
Multiple Choice Question (MCQs)
- The elasticity of highly elastic body is: a) 1 b) 0 c) 0.5 d) ∞ (IOM 2009)
- Breaking stress of a wire of radius 3 mm is F. The breaking stress of the same material of radius 6 mm will be a) F b) 𝐹 2 c)^
𝐹 3 d)^
𝐹 4 (MOE 2012, 2013)
- The stress strain graph is obtained for copper and rubber. Slopes of rubber and cupper are given as tan𝜃𝑟 and tan𝜃𝑐 respectively then:
a) tan𝜃𝑟< tan𝜃𝑐 b) tan𝜃𝑐 ≥ tan𝜃𝑟
c) tan𝜃𝑐 = tan𝜃𝑟 d) tan𝜃𝑐 > tan𝜃𝑟 (MOE 2010)
- The energy density of a wire of strain S is
a) 𝑆^2 2𝑌 b)^
𝑆^2 𝑌 2 c) 2s
(^2) Y d) 2𝑌 𝑆^2 (MOE 2014)
- A metal rod of Young’s modulus of elasticity 2× 10^11 N/m^2 undergoes an elastic strain of 0.05. The energy stored per unit volume in the rod in 𝐽𝑚−3^ is a) 2.5 × 10^8 b) 5 × 10^9 c) 3.5 × 10^7 d) 4 × 10^6 (MOE 2065)
- In order to elongate a given wire to double its length, one requires a stress which is a) of infinite magnitude b) equal to Young’s modulus c) double the Young’s modulus d) half of Young’s modulus (IOM 2009)
- When an elastic material with Young’s modulus Y is subjected to stretching stress S, the elastic energy stored per unit volume of the material is a) 𝑌𝑆 2 b)^
𝑆^2 2𝑌 c)^
𝑆^2 𝑌/2 d)^
𝑆 2𝑌 (MOE 2010)
- Two wires of same radius and material have their length in the ratio 1 : 2, if they are stretched by the same force, then strain produced will be in the ratio of a) 1 : 1 b) 1 : 2 c) 2 : 1 d) 1 : 4 (KU 2017)
- Breaking force for a wire of radius r of given material is F. The breaking force for the wire of same material of radius 2r is a) F b) 2F c) 4F d) 𝐹 4
- Which of the following substances possesses the highest elasticity? a) rubber b) glass c) copper d) steel (BPKIHS 2005)
- A metallic bar is heated from 0 𝑜C to 10 0 𝑜C but it is so held that it can neither expand nor bend, then the force developed is a) F ∝ 𝑙 b) F ∝ 1 𝑙 c)^ F^ ∝ 𝑙
(^0) d) F ∝ 1 𝐴
- A liquid has only a) Shear modulus b) Bulk modulus c) Young’s modulus d) All of the above
- A wire whose cross-section area is A 1 is stretched by 𝑙 1 by a certain weight. How far will a wire of same material and same length and cross-section area A 2 stretch if same weight is applied to it a) 𝐴 1 𝑙 1 𝐴 2^ b)^
𝐴 2 𝑙 1 𝐴 1 c)^ 𝑙^1 d) None of these
- Which type of material is represented by the diagram?
a) perfectly elastic b) partially elastic c) completely plastic d) perfectly rigid
- The ratio of lateral strain to longitudinal strain in a stretched material is called: a) Young’s modulus b) Bulk modulus c) Poisson’s ratio d) Shear modulus
- The Poisson’s ratio for linear elastic material cannot have the value a) 0.2 b) 0.5 c) 0.7 d) 0.
- What happens when a material is stretched beyond its elastic limit? a) It regains its original shape completely. b) It undergoes permanent deformation. c) It becomes more elastic. d) It returns to its original shape faster.
- A wire of length L is cut into two equal parts. How does the Young’s modulus of each half compare to the original wire? \a) It is doubled. b) It is halved. c) It remains the same. d) It becomes four times the original value.
Strain
Fracture Strain^ point
