Unit 10

Mechanical Properties of Matter (Unit 10)

Exam secrets of elasticity, Young's modulus, viscosity, and surface tension.

Core Concepts

• Hooke's Law: Within the limit of proportionality, the extension (e) of a wire is directly proportional to the applied load (F) (F = ke).
• Stokes' Law (F = 6πηrv): The formula for finding the viscous drag force acting on a small spherical object moving through a viscous medium.

Paper Marking Secrets

⚠️ Unit Conversion Trap: In Young's modulus (Y) calculations, the diameter of the wire is often given in millimeters (mm). Students frequently forget to convert it to meters (m), losing all marks! (1 mm = 10⁻³ m).

📘 Resource Book Summary: Mechanical Properties of Matter

Key Definitions:

Elasticity: The ability of a material to deform under an applied force and return to its original shape and size when the force is removed.
Tensile Stress: The tensile force acting per unit cross-sectional area of an object.
Tensile Strain: The extension (change in length) per unit original length of the object.
Young's Modulus (Y): The ratio of tensile stress to tensile strain within the limit of elasticity.

Formulas and Equations:

Hooke's Law: F = ke
Stress: Stress = F / A
Strain: Strain = e / l
Young's Modulus: Y = (F * l) / (A * e)
Energy stored in wire: W = 1/2 * Fe

Key Points:

Strain has no units or dimensions.
On a stress-strain graph, the plot is a straight line from the origin to the proportional limit (Hooke's law is obeyed).

⚠️ Special Notes (Exam Notes/Traps):

Young's modulus (Y) equation applies only within the elastic limit. Moreover, since Young's modulus changes with temperature, the temperature must remain constant during the calculations.

Key Definitions:

Viscosity: The internal frictional force arising between layers of a fluid opposing their relative motion.
Streamline Flow: The orderly, steady flow of fluid particles where each particle passes a point along the same path with the same velocity.
Velocity Gradient: The rate of change of velocity with distance perpendicular to the layers of flow (dv / dx).

Formulas and Equations:

Viscous force: F = η * A * (dv / dx)
Poiseuille's Equation: V / t = (π * a⁴ * Δp) / (8 * η * l)

Key Points:

The coefficient of viscosity (η) of liquids decreases as temperature increases.
When a fluid flows through a tube, the velocity of the layers near the wall is zero, while the velocity of the layer along the axis is maximum.

⚠️ Special Notes (Exam Notes/Traps):

Poiseuille's equation applies only to streamline flow. This means it holds only when the pressure difference is small, the tube is horizontal and narrow, and the fluid is incompressible. Applying it to turbulent flow is incorrect.

Key Definitions:

Stokes' Law: The formula for finding the viscous frictional force acting on a small spherical object moving through a viscous fluid.
Terminal Velocity (v): The constant maximum velocity attained by an object falling through a fluid when the downward gravitational weight is balanced by the upward forces (upthrust + viscous force).

Formulas and Equations:

Stokes' Force: F = 6 * π * η * a * v
Terminal Velocity: v = (2 * a² * (ρ - σ) * g) / (9 * η)

Key Points:

If the density of the sphere is greater than the density of the fluid (ρ > σ), it moves downwards; if less (ρ < σ), it moves upwards (e.g., an air bubble in water).

⚠️ Special Notes (Exam Notes/Traps):

Stokes' law applies only to small, spherical, smooth objects in an infinitely wide stationary fluid. In practical experiments, container wall effects and fluid turbulence can cause errors.

Key Definitions:

Surface Tension (T): The force acting per unit length perpendicular to a hypothetical line drawn on the liquid surface, acting along the surface.
Angle of Contact (θ): The angle measured inside the liquid between the solid surface and the tangent drawn to the liquid surface at the point of contact.
Capillarity: The phenomenon of the rise or fall of liquid level when a narrow tube is dipped in the liquid.

Formulas and Equations:

Surface Tension: T = F / l
Surface Energy: E = T * ΔA
Excess Pressure: Soap bubble: ΔP = 4T/r, Drop/Air bubble: ΔP = 2T/r
Capillary rise: h = (2 * T * cosθ) / (r * ρ * g)

Key Points:

The contact angle between pure water and clean glass is zero (0°). Therefore, cos0° = 1 is used in calculations.
Adding soap or detergent decreases the surface tension of water.

⚠️ Special Notes (Exam Notes/Traps):

Since a soap bubble has two surfaces (inner and outer), you must multiply the surface energy and excess pressure equations by 2. An air bubble has only one surface. In capillary rise calculations (h), measurements must be taken from the bottom of the meniscus.

Physical Quantity	Symbol	SI Unit	Dimensions
Tensile Stress	-	N m⁻² (or Pa)	ML⁻¹ T⁻²
Young's Modulus	Y	N m⁻² (or Pa)	ML⁻¹ T⁻²
Coefficient of Viscosity	η	N s m⁻² (or Pa s)	ML⁻¹ T⁻¹
Surface Tension	T	N m⁻¹	MT⁻²
Force Constant	k	N m⁻¹	MT⁻²

💧 Viscosity Drop Test

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