Elasticity Notes PDF

# Concept of Elasticity PREPARD BY JOSEPH Solids tend to change their shape or size when sufficiently strong external forces are applied to them and to return to their original shape or size after the forces causing the change are removed. Solids which retain their shape or size after the force ca...

# Concept of Elasticity PREPARD BY JOSEPH Solids tend to change their shape or size when sufficiently strong external forces are applied to them and to return to their original shape or size after the forces causing the change are removed. Solids which retain their shape or size after the force causing the change has been removed are said to be 'elastic', and this property of solids is called elasticity. ## Definitions Elasticity is the ability of a substance to regain its original shape and size after being distorted by an external force. An elastic material is one that regains its original shape and size after the distorting external force has been removed. ## Hooke's Law ### Statement of Hooke's Law Hooke's law states that, provided the elastic limit of an elastic material is not exceeded, the extension, *e*, of the material is directly proportional to the applied force, *F*. ### Mathematically, i.e *F α e* <br/> *F = Ke* ...... (1) where *k* is the constant of proportionality called elastic constant or force constant or stiffness of the material. From (1), *K = F/e* ...... (2) If *F* is in newtons and *e* in meters, then *K* is in newtons per meter (Nm^-1) ### Definition: Elastic constant or stiffness of an elastic material is the force required to produce unit extension of the material. The working of the spring balance is based on Hooke's law. In this case *F = mg*, the weight of the body which is proportional to *e*, the extension of the spring. ## Verification or Demonstration of Hooke's Law **Diagram of verification setup:** A spring is suspended from a fixed point with a scale underneath. A scale pan hangs from the end of the spring. A pointer is attached to the spring to track its movement against the scale. **Steps:** 1. Obtain a spiral spring and attach one end of it to a firm support. To the other end attach a scale pan which has been previously weighed. 2. Attach a pointer, *P*, to the spring with plasticize and arrange vertical scale over which the pointer movement can be read. Set up the whole apparatus as shown above. Note the initial pointer reading on the scale and call it *l*. 3. Add weights to the scale pan in equal steps of about 0.05 kg and note and record the pointer reading for each increasing load. Note that the total load stretching the spring in each instance is the weight in the scale pan plus the weight of the scale pan. 4. Remove the load from the scale pan in equal steps also and note the record the corresponding pointer readings. 5. Record your readings as shown in the table below: | Load | Load increasing | Load decreasing | Average reading | Extension | | ----- | -------------- | -------------- | -------------- | --------- | **I. Plot a graph of extension against load. If the total maximum load has not extended the spring permanently, the graph obtained will be a straight line graph passing through the origin as mentioned above. Since the graph is a straight line, it means that the extension of the spring is directly proportional to the load. Hence, Hooke's law is verified** ## Graph of Extension against Load for a Permanently Strained Elastic Material **(a)** Extension | | |--------•B | •Y | •E | •P | |_____O_________Load **(b)** Load | | •E | •P | |_____O_________Extension **Graphs of (a) extension against load (b) load against extension up to breaking point** If, in the Hooke's law experiment, you continue to increase the load until the spring wire snaps or breaks, the graph of extension against load will be as shown in graph above. The graph is a straight line up to *OP*, followed by the curved part *PEYB* which rises slowly initially and then sharply. Along *OP*, the straight portion of the graph, Hooke's law is obeyed and the extension is proportional to the load. *P* is called the 'proportionality limit'. However, up to the point *E* (slightly above *P*) which is called the 'elastic limit'. The spring will return to its original length if the applied force is removed. ### Definition The elastic limit of an elastic material is the limit of force beyond which the stretched material does not return to its original length when the stretching force is removed. Beyond the elastic limit, *E*, the material loses its elasticity. The region *OE* from the origin to the elastic limit *E* is called the 'elastic region'. The difference between the 'proportionality limit' *P* and the 'elastic limit' *E* is that, strictly speaking, Hooke's law holds up to *P*, but elasticity holds up to *E*. Beyond *E*, the wire stretches rapidly with increasing load and then enters the 'elastic region' which begins at *E*. In this 'plastic region' the wire does not return to its original length upon removal of the stretching force. At *Y*, the 'yield point' the material is said to have 'yielded or lost all elasticity permanently and has become 'plastic'. ## Definition Yield point is the point beyond the elastic limit in which the elastic material has yielded all its elasticity permanently and has become plastic. The difference between the “elastic region" (*OE*) and the "plastic region" (*EB*) is that in a latter, a sudden and rapid increase in the extension for any slight increase in load. *B* is the 'breaking point' where the material may finally 'break' or 'snap'. It corresponds to the maximum possible extension that can be produced in the material. ## Examples 1. A force of 10N is required to extend the length of a spiral spring by 4 cm. Find the extension of the spring when a mass of 360 g is suspended from its end (g = 10ms^-2). ### Solution By Hook's law, *F = ke* <br/> .. *k = F/e* <br/> = 10N/0.04 m When a body is suspended, the weight *w* should produce an extension, *e* <br/> .. *W = mg = kxe* <br/> ... *e = mg/k* <br/> = 360 kg x 10 N / 1000 kg x 4/1000 N m <br/> = 1.44 x 10^-2 m <br/> = 1.44 cm. ## Examples 2. An elastic material is stretched 45 mm by a force of 3 N. Calculate the additional force which will stretch the material 50mm, if the elastic limit is not exceeded. ### Solution By Hook's law, <br/> *F = k.e* <br/> .. *k = F/e = 3 N / 45 x 10^-3 m* Let the force stretching the material 50 mm be *F* <br/> ... *F = k.e* <br/> = k x 50 x 10^-3 m <br/> = 3N / 45 x 10^-3 m x 50 x 10^-3 m <br/> = 3.33 N Additional force <br/> = (3.33 - 3.00) N <br/> = 0.33 N ## Young's Modulus of Elasticity Suppose a wire of length *l* (m) and cross-sectional area *A* (m²) is extended through *e* (m) by a force *F* (N). (a) The ratio of the force to the area, *F/A* is called the stress or 'tensile' of the elastic material. <br/> Stress = *F/A* ...... (1) (b) The ratio of the extension, *e* to the original length, *l* of the wire i.e (*e/l*) is called the tensile strain of the wire. <br/> ... Strain = *e/l* ...... (2) Hence Hooke's law can also be stated as follows: The tensile stress of the material is directly proportional to its tensile strain provided the elastic limit is not exceeded. ## YOUNG MODULUS It is the constant of proportionally when stress is directly proportional to strain. <br/> Young's modulus = Stress/Strain <br/> Y = *FA/l* The unit of *Y* is Nm^-2 (Newton per square metre) the same unit as stress, since strain has no unit. ## 3.0 Work Done in Springs and Elastic String Diagram of a spring: A spring has a natural length. When a force is applied to it, the spring either compresses and shortens, or extends and lengthens. **Work done in extending or compressing a spring** Consider a spring of unstretched length, *l* (m). Suppose a force *F* (N) causes an extension (or compression) *e* (m). The force in the wire has increased from an initial value of zero at *P* to a value of *F* at 'P'. The work (*W*) done in stretching the spring through the extension is given by: *W* = Average force x extension <br/> = (Initial force + final force)/2 x extension <br/> ... *W* = 0 + *Fxe*/2 <br/> ... *W* = 1/2 *Fe* .......... (1) But *F = ke* (Hooke's law) <br/> ... *W* = 1/2 *ke²* .........(2) This work done is stored as potential energy in the material which can do work when the stretching force is removed and the material regains its original length. This type of stored energy is applied in the working of many spring clocks and toys. The energy is stored in the clock spring (or toy spring) which is compressed by winding. ## Elastic Potential Energy ### Definition The elastic potential energy of a stretched or compressed material is the ability of the material to do work. Elastic potential energy arises because work is done in stretching or compressing the material as we have seen in the previous section <br/> *W* = 1/2 *Fe* = 1/2 *ke²* Where *F* is the maximum stretching (or compressing) force, *e* is the extension (or compression) and *k* is the force constant or stiffness of the material. ### Example (or application) of elastic potential energy: When you stretch the rubber of a catapult and project a stone, the elastic potential energy stored in the rubber is converted into the kinetic energy of the flying stone according to the law of conservation of energy. ## Examples 1. An elastic material is stretch 1.2 cm by a mass of 96g suspended from one end of it. What is the elastic constant of the material? ### Solution Weight of object = stretching force <br/> = 960 x 10 N/1000 (W = mg) <br/> = 9.6 N <br/> Extension = *F/ke* <br/> = 9.6 N / 1.2 x10^-2 m <br/> .. *k* = 9.6 / 1.2 x10^-2 Nm^-1 <br/> = 800 Nm^-1 ## BULK MODULAS It is the measure of how compressible that substance is. It is the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. The inverse of the bulk modulus give a substances compressibility. For fluid stress, only bulk modulus is meaningful, Young modulus deals with linear stress in solid. Bulk Modulus = volume of Stress / Volume of strain ## SHEAR MODULUS Shear modulus is the ratio of shear stress to shear strain. The shear modulus is also known as the rigidity in metal. It is the numerical constant that describe the elastic properties of solid. Shear modulus= shear stress / shear strain <br/> Measured in N/M² ## APPLICATION OF ELASTICITY 1. Property of elasticity is used in sliver dressing in wound care 2. It is used for the treatment of patient with venous disease 3. Blood vessels are elastic in nature 4. Elastic materials are used in PPE (personal protective equipment such as gloves, masks and gown to prevent exposure to bodily fluids. 5. Elastic casting material are used to create flexible casts and splints that allow for movement and flexibility. 6. Elastic material are used in medical devices such as stethoscope, blood pressure cuffs. 7. Elastic bandage and dressings are used to provide compression and support to wounds, promoting healing and tissue elasticity.