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PROPERTIES OF MATTER Stress is defined as Restoring force per unit area. A stress may be (i) tensile, (ii) compressible, (iii) shearing or tangential, and (iv) hydrostatic. These are illustrated in the figures below : All stresses are measured in units of Nm-2. The dimensions are the same as those of pressure viz., ML-1T-2. See the picture Strain = Linear strain = l / l Shear strain = ߠ = x / l Volume strain = - V / V All’s strain’s are dimensionless and are pure numbers without any units. = Constant = Modulus of electricity The relation is known as Hooks Law Young’s Modulus Y = Bulk Modulus K = -V Shear Modulus or Modulus or Rigidity = Poisson’s Ratio = = Pascal’s Law : The pressure exerted at any point in an enclosed fluid is transmitted equally in all directions. Archimede’s Principle : When a solid body is wholly or partially immersed in a fluid, it experiences an upward buoyant force which acts through its centre of buoyancy. Law of Flotation : When a body is just floating in a fluid, the weight of the fluid displaced by it equal the total weight of the body and the centre of gravity of the body and the centre of buoyancy lie along the same vertical line. Motion in a Vertical circle : Let a body tied to one end of a string be whirled in a vertical circle of radius r whose other end O is fixed. Let at any time t, the body be at position P where the string makes an angle ߠ with the vertical. If T is the tension in the string. Tension in string T = + Mg cosߠ At the lowest point A, ߠ = 0 tension at point A will be TA = + Mg At the highest point B, ߠ = , for highest point B it will be TB = – Mg If TB is greater than zero, the string remains tight, but when TB < 0 or negative, the string becomes lose and the body no longer remains on circular path. For just completing the vertical circle TB = 0, then the speed at highest point B is called the critical speed Vc. Vc = This equation gives critical velocity at highest point. If the velocity VB < VC, the string becomes slack and the body falls downward instead of moving along the circular path. Case (i) Condition for performing complete revolution : The condition for the body to perform complete revolution is that tension in the string must be always greater than zero at the highest point. For this the velocity at highest point > critical velocity, VC = and the velocity at lowest point VA > (5rg). Case (ii) Condition of Oscillation If velocity of lowest point VA = , The particle will oscillate about the lowest point A. Case (iii) Condition for the body to leave the circular path. The body leaves the circular path when the tension in string becomes zero earlier than the velocity VA > Thus if the speed of body at lowest point VA lies between and , the body leaves the circular path somewhere between A and B. Concial Pendulum It consists of a string OA, whose upper end O is fixed and a bob is tied at the other free end. The bob is given a horizontal push a little through angular displacement ߠ arranged such that the bob describes a horizontal circle with uniform angular velocity in such a way that the string always makes an angle ߠ with the vertical. As the string traces the surface of the cone the arrangement is called a conical pendulum. Surface Tension : It is measured as the tangential force per unit length acting along any line drawn on the liquid surface. It also equals the free energy per unit area. T= = The dimensions of ‘T’ are [MT-2] and its units are Nm-1 or Jm-2. Excess pressure : for a drop : p = 2T/r Excess pressure : for a Soap bubble = p = 4T/r. (Because it contains two surfaces) When a tube of radius (r) is immersed in a liquid of surface tension (T) and height of rise of liquid of density ( ) and intact angle (ߠ). Then for << h, we have T= For tubes of ‘insufficient length’ = = = a constant Reynold’s Number N= = a dimensionless quantity Equation of Continuity : For liquids flowing in tubes of area of cross section A & velocity v, then A = constant. Bernoulli’s Theorem. For a liquid undergoing stream-lined flow. Kinetic energy + Gravitational potential energy + Pressure energy = Constant. or + gh + P = constant Torricelli’s Theorem. The liquid placed in a container to a height having a hole at the bottom then Velocity of efflux = gh. It is independent of liquid. Newton’s Law of Viscosity. The force required to maintain flow is F=- A where is coefficient of viscosity, A area of cross-section of the tube, is the velocity gradient between two layers of a liquid. are ML-1T-1 and its SI unit is Tor. This is related to its more often The dimensions of used unit, the poise, by the relation 1 Tor = 10 poise When a particle of radius r placed in a liquid of coefficient of vislosity v moving with a constant velocity after some time. Then the force experienced by the particle is F= r The volume of liquid flowing in tubes is given by Volume rate of flow = V = (Poiseullie’s law)