Mechanical Properties of Solids
Stress and Strain
Stress: Force per unit area. sigma = F/A. SI unit: Pa (N/m^2).
Strain: Ratio of change in dimension to original dimension. Dimensionless.
Types of Stress:
- Tensile stress: Pulling force.
- Compressive stress: Pushing force.
- Shear stress: Tangential force.
Types of Strain:
- Longitudinal strain:
Delta L/L - Volumetric strain:
Delta V/V - Shear strain:
theta = x/L
Hooke's Law
Within elastic limit, stress is proportional to strain.
stress/ /strain = E (Modulus of Elasticity)
Elastic Moduli
Young's Modulus (Y): Y = (F/A)/(Delta L/L) = (FL)/(A Delta L)
For stretching/compression of rods and wires.
Bulk Modulus (B): B = - (Delta P)/(Delta V/V) = -V Delta P/Delta V
For volume change under pressure.
Compressibility = 1/B.
Shear Modulus (G): G = (F/A)/(x/L) = (FL)/(A x)
For shape change under shear stress.
Poisson's Ratio
sigma = text(lateral strain)/text(longitudinal strain) = (Delta d/d)/(Delta L/L)
Elastic Limit and Yield Strength
Beyond the elastic limit, permanent deformation occurs. The maximum stress a material can withstand without permanent deformation is its yield strength.
Mechanical Properties of Fluids
Pressure in Fluids
P = F/A. SI unit: Pa.
Hydrostatic Pressure: P = P_0 + rho g h (at depth h).
Atmospheric Pressure: 1 atm = 1.013 x 10^5 Pa.
Pascals Law
Pressure applied to an enclosed fluid is transmitted equally to every point in the fluid and to the walls of the container.
Applications: Hydraulic lift, hydraulic brakes, hydraulic press.
F_1/A_1 = F_2/A_2 => F_2 = F_1 (A_2/A_1) (force multiplication).
Archimedes Principle
When a body is partially or fully immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced.
F_b = rho_fluid * V_imm * g
Law of Floatation
A body floats if its weight equals the weight of fluid displaced (buoyant force).
rho_body < rho_fluid => floats
rho_body = rho_fluid => hangs suspended
rho_body > rho_fluid => sinks
Bernoullis Theorem
For an incompressible, non-viscous fluid in steady flow:
P + (1/2)rho v^2 + rho g h = constant
Applications
- Venturimeter: Measures flow speed.
v = sqrt(2(P_1-P_2)/(rho(A_1^2/A_2^2 - 1))). - Atomiser: Higher air speed creates lower pressure, drawing liquid up.
- Aerofoil lift: Shape causes faster airflow on top, creating lower pressure (lift).
- Magnus effect: Spinning ball curves due to pressure difference.
Surface Tension
The property of a liquid surface to behave like a stretched membrane.
S = F/L (force per unit length). SI unit: N/m.
Surface Energy
U = S * Delta A (work done per unit increase in area).
Capillary Rise
h = (2S cos theta)/(rho g r)
Factors Affecting Surface Tension
- Temperature: Surface tension decreases with increasing temperature.
- Impurities: Some increase, some decrease surface tension.
Viscosity
Internal friction in fluids. Measured by coefficient of viscosity eta.
Newtons Law of Viscosity
F = -eta A (dv/dx) where dv/dx is velocity gradient.
Stokes Law
Viscous force on a sphere: F = 6pi eta r v (for laminar flow).
Terminal Velocity
v_t = (2r^2 (rho - sigma)g)/(9eta)
Where rho = density of sphere, sigma = density of fluid.
Worked Examples
Example 1: A wire of length 2 m and cross-section 10^(-6) m^2 is stretched by 1 mm under a 50 N load. Find Young's modulus.
Solution: Y = (F/A)/(Delta L/L) = (50/10^(-6))/(0.001/2) = 5x10^7/5x10^(-4) = 1x10^11 Pa.
Example 2: Water rises to 5 cm in a capillary tube of radius 0.5 mm. Find surface tension (density = 1000 kg/m^3, g = 10 m/s^2, cos theta = 1).
Solution: S = (h rho g r)/(2 cos theta) = (0.05*1000*10*5x10^(-4))/2 = 0.125 N/m.
Common Mistakes
- Elastic limit vs plasticity: Elastic deformation is reversible; plastic deformation is not.
- Pascals law limitation: Only true for enclosed fluids, not open containers.
- Bernoulli applies only for non-viscous flow: Viscous fluids with turbulence violate the theorem.
- Capillary rise formula: Depends on
cos theta. For non-wetting liquids (mercury),theta > 90=> depression.
ISC Exam Focus
- Theory (70%): Stress-strain, elastic moduli, Pascal's law, Bernoulli's theorem, surface tension.
- Application (30%): Numerical problems on Young's modulus, capillary rise, terminal velocity.
- ISC frequently asks: "Derive expression for capillary rise" or "State and prove Bernoulli's theorem."
- Combined numericals on stress-strain and fluid properties.
Self-Test Questions
Q1: Define Young's modulus. Write its SI unit.
Answer: Y = FL/(A Delta L). SI unit: N/m^2 or Pa.
Q2: State Pascal's law. Give one application. Answer: Pressure applied to enclosed fluid is transmitted equally. Application: hydraulic lift.
Q3: In a hydraulic lift, the area ratio is 100:1. What force is needed to lift a 2000 kg car?
Answer: F_1 = F_2 * A_1/A_2 = 2000*10 * 1/100 = 200 N.
Q4: State Bernoulli's theorem for an ideal fluid.
Answer: P + (1/2)rho v^2 + rho g h = constant for incompressible, non-viscous, steady flow.
Q5: A sphere of radius 2 mm falls in a liquid of viscosity 0.8 Pa s with terminal velocity 0.1 m/s. Find viscous force.
Answer: F = 6pi eta r v = 6*3.14*0.8*0.002*0.1 = 0.00301 N.
Q6: State the law of floatation. Answer: A body floats if its weight equals the weight of fluid displaced, i.e., the buoyant force equals the weight of the body.
