UNIT-1 PROPERTIES OF MATTER PDF

UNIT- 1 PROPERTIES OF MATTER Introduction to Elasticity - Stress Strain, Hooke‟s Law - Torque, Moment of Force & Couple - Stress-Strain diagram Torsion pendulum - Experiment to determine Rigidity modulus - Bending o...

UNIT- 1 PROPERTIES OF MATTER Introduction to Elasticity - Stress Strain, Hooke‟s Law - Torque, Moment of Force & Couple - Stress-Strain diagram Torsion pendulum - Experiment to determine Rigidity modulus - Bending of beams - bending moment, Cantilever - Derivation of bending moment, Experiment to determine Young‟s Modulus - uniform and non-uniform bending 9 Hrs. 1.1 Introduction to Elasticity The advancement in Technology & Engineering is possible through our detailed understanding of properties of matter. The mechanical properties of solids generally deals with deformations due to the external forces impressed on the materials. The property, elasticity is important for many applications in Engineering & Technology.[e.g a bridge used for traffic is subjected to loads or forces of varying amounts. Before a steel bridge is constructed, the steel samples are sent for testing to find whether the steel can withstand the loads likely to be put on them or not]. The development of dynamics begins with the concept of an infinitely small but massive particle for their behaviour under the influence of applied forces. Later the idea of rigid body came into force. The behaviour of ordinary material under the action of forces constitutes the study of elasticity. A body can be deformed (i.e. change in shape or size) by applying a suitable force on it. After removing the force, if the body regains its original shape and size, then it is perfectly elastic body. Definition The property on account of which, the body opposes the deforming forces and regains its original shape and size on the removal of the deforming forces is called elasticity. Elastic Body: If a body completely regains its original shape and size on the removal of the deforming forces, the body is said to be perfectly elastic. This property of the body is called elasticity. Examples: Rubber and Metal Deformation Force: The forces which changes or tries to change the shape or size of a body without moving it as whole is called deforming force. 1.2 STRESS When external forces acting on a body produce some deformation of the body, then the forces of reaction is set with in the body to restore the body to its original shape and size. The restoring force acting per unit area inside the body is defined as stress. The restoring force is equal and opposite to the applied force F, therefore, stress is also defined as “The deforming force per unit cross sectional area of body” Stress = More appropriate definition of stress is Dimension: [ML−1T−2] Unit: Newton/m2 There are three types of stress (a). Linear or Longitudinal Stress (b). Shearing or Tangential Stress (c). Volume or Bulk Stress 1.2 a. Longitudinal stress (or) Tensile stress  If the deforming force acting on a body is along its longitudinal axis and produces a change in its length, then the deforming force / unit area acting normal to the surface is called longitudinal or normal or tensile stress. If the applied force is a thrust or produces a compression, then the stress is called normal compressive stress. Fig 1.1 Longitudinal stress Thus, Tensile stress = force / unit area =F/A Tensile stress = F=A 1.2 b. Shearing stress (or) Tangential stress: If the deforming forces are applied tangentially over the top surface of a cube and bottom surface being kept fixed then, the top face gets displaced towards the direction of applied force. Fig 1.2 Shearing stress The tangential force/unit area or the stress which tends to make one part of the body slide across the other part is termed as shearing stress or tangential stress. 1.2 c. Volume or Bulk stress If the equal deforming forces can be applied uniformly on each 6 faces of a cube in outward direction, then the cube suffers an increase in its volume. Under equilibrium, the applied force/unit area is called volume or bulk stress. Fig 1.3 Volume stress 1.3 STRAIN External forces acting on a body produce a change in its shape and size. The fractional change is called strain. The change in dimension or shape of a body due to the deforming force results in strain. The strain is measured by the ratio of change in dimension to original dimension Types of Strains: According to the change in length or area or volume, there are three types of strains namely (a)Linear strain (Change in length per unit length )  (b) Shearing strain (Change in area per unit area) (c) Volume or bulk strain (Change in volume per unit volume) 1.3 a Longitudinal Strain: Definition: It is defined as the ratio between the changes in length to original length, without any change in its shape, after the removal of external forces. Fig 1.4 Longitudinal strain 1.3 b. Shearing Strain Definition: It is defined as the angular deformation produced on the body due to the application of external tangential forces on it.  Fig 1.5 Shearing strain 1.3 c. Volume or bulk strain Definition: It is defined as the ratio between changes in volume to the original volume, without any change in its shape Fig 1.6 Volume strain Elastic limit The maximum stress upto which a body exhibits the property of elasticity is called elastic limit or limit of elasticity. 1.4 HOOKE’S LAW Robert Hooke an English Physicist in the year 1679 had given a relation between stress and strain. This relation is known as Hooke‟s law. Statement: It states that within the elastic limit stress in a body is directly proportional to the strain produced on it. In other words the ratio between stress and strain is a constant. This constant of proportionality is known as coefficient of elasticity or modulus of elasticity. It has the units of stress i.e. N/m 2 and dimensions are [ML−1T−2] 1.5 STRESS –STRAIN DIAGRAM Consider a wire which is rigidly fixed at one end. It is loaded at the other end. The strain produced for the different loads are noted until the wire breaks down. This graph is known as stress-Strain diagram or graph. The following useful information regarding the behavior of solid materials is studied from this diagram. Fig 1.7 Stress and Strain diagram 1.5 a. Hooke’s Law The portion OA of the curve is a straight line. In this region, stress is directly proportional to strain. This means that upto OA, the material obeys Hooke‟s law. The wire is perfectly elastic, The point A is called the limit of proportionality. 1.5 b. Elastic limit The stress is further increased till a point A′. The point A′, lying near to A denotes the elastic limit. Upto the point A′ the wire regains its original length if the stress is removed. If the wire is loaded beyond the elastic limit, then it will not return its original length. 1.5 c. Yield point On further increasing the stress beyond the elastic limit, the curve bends and a point B is reached.In this region A′B, a slight increase in stress produces a larger strain in the material. The point B is called yield point. The value of stress at the point is called yield strength of the material. 1.5 d. Permanent set In the region A′ B, if stress is removed, the wire will never return to its original length. The wire is taken a permanent set. 1.5 e. Plastic range Beyond B the strain in the wire increases rapidly without any increase in the load. This is known as plastic range. 1.5 f. Ultimate tensile strength If the wire is further loaded, appoint C is reached after which the wire begins to neck down. Hence its cross sectional area is no longer remains uniform. At the point C, the wire begins to thin down at the same point and it finally breaks. At the point C, the stress developed is maximum and it is called Ultimate tensile strength 1.5 g. Breaking point. The point „B‟ is known as the breaking point where the wire breaks down completely. The stress at the point D is called breaking point. 1.6 MOMENT, COUPLE AND TORQUE 1.6 a. Moment of a force: The moment of a force about a point is defined as the product of the magnitude of the force and perpendicular distance from the point to the line of action of force. Fig 1.8 Moment of Force Explanation: Let “F” be the force acting on a body, at A as shown in figure. Then, the moment of force F about “O” is. Where d is the perpendicular distance from the point to the line of action of force F. 1.6 b. Couple: A couple constitutes a pair of two equal and opposite force acting on a body, in such a way that the lines of action of the two forces are not in the same straight line.  Fig 1.9 Couple Explanation: These two forces form a couple and the moment of the couple about A is MA and about B is MB, then we can write. 1.6 c Torque: Torque is the rotating force and is equal to the moment of the couple. Torque is the product of one of the forces forming couple and perpendicular distance between the two opposite forces. Fig 1.10 Torque Explanation: Let us consider two equal and opposite forces F and -F acting tangentially on the rim of the circular disc from the extremities of diameter as shown in figure. If forms a couple. If the rotates through a small angle θ, the distance moved by the force F in rotating the body through an angle θ= length of the arc AB=θ  Here the length of the arc AB= r θ Work done by a single force =F r θ Work done by two forces constituting a couple = 2 F r θ Here 2 F r θ is the moment of the couple (or) torque τ Work done by the torque = τ θ 1.7 TORSION PENDULUM (Determination of Moment of Inertia, Rigidity Modulus and Torque per unit twist) Theory A circular metallic disc suspended using a thin wire that executes torsional oscillation is called torsion pendulum. Principle: When a disc (torsion pendulum) is rotated in a horizontal plane, the disc executes simple harmonic oscillation due to the restoring couple produced in the wire. It executes torsional oscillations whereas a simple pendulum executes liner oscillations Description A torsion pendulum consists of a wire with one end fixed to a split chuck and the other end fixed to the centre of the circular disc of radius R as shown in fig. Let L be the distance between the chuck end to the disc and r be the radius of the wire. Fig 1.11 Torsional Pendulum  Fig 1.12 Without Mass With Mass at Close distance With Mass at farthest distance Working When the suspension wire is twisted by the circular disc fixed at the bottom of the wire, the wire undergoes shearing strain. This is called torsion. Because of this torsion, the disc executer oscillations called torsional oscillations. The torsional pendulum consists of a circular disc suspended by a thin suspended wire, as shown in fig., whose rigidity modulus is to be determined. The top end of the wire is fixed by a chuck. The circular disc is attached to the other end of the wire. (i) Calculation of To: Adjust the wire so that its length is fixed value say 60cms or 70cms. Makea vertical chalk mark on the disc when it is rest as a reference. By making a small twist to the circular disc, setup Torsional oscillations. After the first few oscillation,just as the mark on the disc passes the equilibrium positions, a stop clock is stated. The time taken for 20 complete oscillations is noted. The experiment is repeated for second trial and means value was calculated. The mean value of the time period is To without any masses on the circular disc. (ii) Calculation of T1: The two identical cylindrical masses are place at equal distance of either side of the central chuck as close as possible. The distance d1 is measured between the wire and the centre of the cylinder. By twisting the disc, the time taken for 20 complete oscillations was noted. The mean value of the time period is called T1. (iii) Calculation of T2: The identical masses are arranged symmetrically as far away from the axis of the rotation as possible. The distance d2 is measured between the centre of the mass oft the disc and the suspension of the wire. The time period T2 is calculated for the time taken for 20 complete oscillations in the same manner as that of the calculation of To and T1. The radius of the wire (r) can be measured by using screw gauge. By doing this experiment the calculable parameters are  Couple per unit twist, c = πnr4 / 2l N-m. By using these formulas I, n and c can be calculated in Torsional Pendulum. 1.8 BENDING MOMENT (BENDING OF BEAMS) Beam A beam is defined as a rod (or) bar of uniform cross section whose length is very much greater than its other dimensions, such as breadth and thickness. It is commonly used in the construction of bridges to support roofs of the buildings etc., Since the length of the beam is much greater than its other dimensions the shearing stresses are very small. Assumptions: (i) The length of the beam should be large. (ii) The load applied should be large. (iii) The cross section of the beam remains constant. (iv) The shearing stresses are negligible. (v) The curvature of the beam is very small. 1.8.a BENDING OF A BEAM AND NEUTRAL AXIS Let us consider a beam of uniform rectangular cross section. This beam is made up of a large number of thin plane layers one above the other. Consider a beam which is bent an arc by the application of a load.  Taking a longitudinal section ABCD of the bent beam, the layers in the upper half are elongated while those in the lower half are compressed. Fig 1.13 Bending of beams-Neutral axis Let us consider a filament AB at the centre of the beam, which is remain unaltered is taken as reference axis called as neutral axis and the plane is called as neutral plane. 1.9 b. EXPRESSION FOR THE BENDING MOMENT OF BEAM Let us consider a beam under the action of deforming forces. The beam bends into a circular arc. Let AB be the neutral axis of the beam. The filament AB remains unchanged. Let PQ be the arc chosen from the neutral axis. If R --- radius of curvature θ --- Angle Fig 1.14 Bending Moment of a beam 1.10 CANTILEVER Cantilever A cantilever is a beam fixed horizontally at one end and loaded at the other end. This type of bending is known as cantilever. Theory Let us consider a beam at one-end and loaded at its other free end. Due to the load applied at the free end, a couple is created between the two forces. (i.e)  Fig 1.15 Cantilever (i). Force (load „W‟) applied at the free end towards downward direction and (ii). Reaction (R) acting in the upward direction at the supporting end. 1.11 DERIVATION FOR DEPRESSION OF A CANTILEVER - LOADED AT ITS ENDS Theory Let „l‟ be the length of the cantilever AB fixed at „A‟. Let „W‟ be the weight suspended (loaded) at the free end of the cantilever. Due to the load applied the cantilever moves to a new position „BB!‟ Fig 1.16 Depression at the free end of the Cantilever Let us consider the section of cantilever P at a distance x from the fixed end A. It is a distance (l-x) from the loaded end B!  The external bending moment = 𝑊 × 𝑃𝐵! = 𝑊 × ( − 𝑥) − −→ (1) We know the internal bending moment = − −→ External bending moment = Internal bending moment 1.12 EXPERIMENTAL DETERMINATION OF YOUNG’S MODULUS BY UNIFORM BENDING METHOD When a beam is loaded at the both end of the beam from the equal distance „a‟ from the knife edge, the beam would form an arc of a circle. This type of bending is called uniform bending. The given beam is placed over the knife edges A and B at a particular distance say 50cm or 60cm in the same horizontal levels. The hanger is placed both end of the beam at the equal distance ‘a’ from the knife edges. At the centre of the bar, a pin is fixed using wax as  shown in fig. A travelling microscope is placed in front of the whole setup for finding the position of the pin. Fig 1.17 Young’s Modulus- Uniform bending Taking the weight hanger alone as the dead load, the midpoint of the pin is focused by the microscope, and is adjusted in such a way that the tip of the pin touches with the horizontal crosswire. The reading on the vertical scale is noted. Now the weight is added in the steps of 50 g on both hangers. Each time, the tip of the pin is made to tough the horizontal cross wire and the readings are noted from the vertical scale of the microscope. By unloading the weight in steps of same 50g the readings are tabulated in table.  To determine Elevation (y): The thickness (d) and the breadth (b) of the beam are measured using the screw gauge and vernier caliper respectively. Substituting the values in equation, Young‟s modulus can be calculated by the relation 1.13 NON UNIFORM BENDING - DEPRESSION AT THE MID POINT OF A BEAM LOADED AT THE MIDDLE. Principle: The Young‟s modulus of elasticity of the beam can be calculated by loading the beam at middle. Hence the depression occurs over beam. Description Let as consider a beam of length l supported on the two knife edges A and B as shown in figure. The load of weight W is suspended at the centre C. It is found that the beam bends and the maximum displacement is at the point D. Due to the load (W) applied at the middle of the beam the reaction 𝖶 is 2 acted vertically upwards at each knife edges. The bending is called as non- uniform bending. The beam may be considered as two cantilevers, whose free carriers a load w / 2 each of the length l/2 and fixed at the point D. Procedure: The given beam is placed on the two knife edges. The length of the bar between the knife edges is measured (l). A weight hanger is suspended at the centre of the beam and a pin is fixed vertically on the frame of the hanger as shown in Fig. A microscope is focused on the tipoff the pin. Taking the weight hanger alone as the dead load the tip the pin is focused by the microscope, and is adjusted in such a way that tip of the pin just touches the horizontal cross wire. Fig 1.18 Young’s Modulus- Non Uniform bending The reading on the vertical scale is noted. Now the weight is added in steps of 50grams. Each time the tip of the pin is made to touch the horizontal cross wire and the readings are noted from the vertical scale of the microscope. The same procedure is repeated by unloading the weight in steps of same 50grams and the readings are noted. To determine Depression (y): Microscope Readings Mean depression y Sl.No Load (M) Increasing Decreasing Mean for a load of M Load Load Kg Unit 10−3Kg 10− m 10− m 10− m 10− m W 1 W+50 2 W+100 3 W+150 4 W+200 5 W+250 Mean y The thickness and the breadth of the beam are measured using screw gauge and vernier calipers respectively. The Young‟s modulus of the material of the beam can be calculated using the formula. 𝑀g 𝑌3 = 𝑁 −2 or 4𝑏𝑑3𝑦 the depression of the beam is 𝑀𝑔 3 𝑦= 4𝑏𝑑3𝑌 Table 1.3: To determine Depression (y): Microscope Readings Mean depression y Sl.No Load (M) Increasing Decreasing Mean for a load of M Load Load Kg Unit 10−3Kg 10− m 10− m 10− m 10− m W 1 W+50 2 W+100 3 W+150 4 W+200 5 W+250 Mean y The thickness and the breadth of the beam are measured using screw gauge and vernier calipers respectively. The Young‟s modulus of the material of the beam can be calculated using the formula. 𝑀g 3 𝑌= 𝑁 −2 or the depression of the beam is 4𝑏𝑑3𝑦  𝑀𝑔 3 𝑦= 4𝑏𝑑3𝑌