Physics
Some Mechanical Properties of Matter
Elasticity
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Elasticity is the ability of a body to return to its original shape after deformation. Stress is the internal force per unit area within a deformed body.
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- Rigid Solid Body (Ideal): Distance between any two particles is always fixed.
- Real Solid Body: Deforms when external forces are applied. Internal forces develop to restore the original shape.
- Elasticity: The property of a material to restore its natural shape or oppose deformation.
- Perfectly Elastic Body: Completely regains its natural shape after removal of deforming forces.
- Perfectly Inelastic/Plastic Body: Remains in the deformed state after removal of deforming forces.
- Partially Elastic Body: Partially regains the original shape.
Microscopic Reason of Elasticity
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Solids are made of molecules in stable equilibrium. Deformation displaces molecules, causing restoring forces to return them to their original positions.
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- Solid composed of molecules/atoms arranged in a particular fashion.
- Each molecule acted upon by forces from neighboring molecules.
- Solid takes a shape where each molecule is in stable equilibrium.
- Deformation displaces molecules from equilibrium. Intermolecular distances change.
- Restoring forces act on the molecules, driving them back to their original positions.
Stress: Longitudinal and Shearing
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Stress is the force per unit area within a material. Longitudinal stress is normal to the area, while shearing stress is tangential.
['Normal Stress/Longitudinal Stress: Γ = Fn / ΔS', 'Tangential Stress/Shearing Stress: Γt = Ft / ΔS']
- Stress: Internal restoring force acting per unit area of a deformed body.
- Consider a body under several forces, with the net force being zero. The center of mass remains at rest. The body deforms and internal forces appear.
- Cross-sectional area ΔS: Parts of the body on either side exert forces F and -F on each other.
Tensile and Compressive Stress
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Tensile stress occurs when a body is stretched, while compressive stress occurs when a body is compressed.
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- Tensile Stress: The two parts of the body on either side of ΔS pull each other. Occurs when a rod or wire is stretched by equal and opposite forces. Fn is the tension.
- Compressive Stress: The rod is pushed at both ends with equal and opposite forces. The two parts of the body on either side of ΔS push each other.
Tensile Stress, Tensile Strain, Young's Modulus
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When a force is applied to stretch a material, it experiences tensile stress and strain. Young's modulus relates these two properties.
['Y = (F/A) / (ΔL/L) = (F * L) / (A * ΔL)']
Tensile stress (F/A) is the force per unit area applied to stretch the material. Tensile strain (ΔL/L) is the fractional change in length due to the applied force. Young's modulus (Y) is the ratio of tensile stress to tensile strain.
Compressive Stress and Strain
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When a force is applied to compress a material, it experiences compressive stress and strain. Young's modulus also applies here.
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Compressive stress is similar to tensile stress but applied to compress the material. Compressive strain is the fractional change in length due to compression. Young's modulus is the ratio of compressive stress to compressive strain and is the same as in the tensile case for many materials.
Poisson's Ratio
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When a material is stretched, it becomes thinner. Poisson's ratio describes how much the width changes relative to the length.
['σ = - (Δd/d) / (ΔL/L)']
Poisson's ratio (σ) is the ratio of the fractional change in transverse length (Δd/d) to the fractional change in longitudinal length (ΔL/L). The negative sign ensures that σ is positive.
Shear Modulus (Modulus of Rigidity/Torsional Modulus)
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Shear modulus describes a material's resistance to deformation when a force is applied parallel to a surface.
['η = (F/A) / (x/h) = (F * h) / (A * x)']
Shear modulus (η) is the ratio of shearing stress (F/A) to shearing strain (x/h), where x is the displacement and h is the height.
Bulk Modulus
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Bulk modulus describes a material's resistance to compression under pressure.
['B = - P / (ΔV/V)', 'B = - ΔP / (ΔV/V) = - V (dP/dV)']
Bulk modulus (B) is the ratio of volume stress (P, pressure) to volume strain (ΔV/V). The negative sign ensures that B is positive since volume decreases with increasing pressure.
Compressibility
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Compressibility is the inverse of bulk modulus and describes how easily a material's volume changes under pressure.
['K = 1/B', 'K = - (1/V) (dV/dP)']
Compressibility (K) is the reciprocal of the bulk modulus (B).
Elastic Potential Energy of a Stretched Wire
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When a wire is stretched, it stores energy due to the work done in stretching it. This stored energy is called elastic potential energy.
['Potential energy = (1/2) * stress * strain * volume', 'U = (AY/2L) * l^2']
When a wire of length L and cross-sectional area A is stretched by a length l due to an external force, the elastic potential energy (U) stored in the wire is given by:
The force is adjusted so that the wire is stretched slowly. At any time during the extension, the external force equals the tension in the wire. If the extension is *x*, the wire is under longitudinal stress *F/A*, where F is the tension at this time. The strain is *x/L*. If Young's modulus is Y, then *F/A = Y(x/L)* or *F = (AY/L)x*.
The work done by the external force in a further extension *dx* is *dW = F dx = (AY/L)x dx*. The total work done in an extension from 0 to *l* is *W = ∫dW = ∫(AY/L)x dx = (AY/2L)l²*.
This work is stored in the wire as elastic potential energy.
U = (AY/2L)l²
This can also be expressed as:
U = 1/2 * (maximum stretching force) * (extension)
U = 1/2 * (stress) * (strain) * (volume)
Determination of Young Modulus in Laboratory
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A method to experimentally determine Young's modulus involves suspending two wires (a reference and an experimental wire) of nearly equal length from a fixed support. A scale is attached to the reference wire, and a vernier scale is attached to the experimental wire to measure the extension under applied load.
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A long wire A (say 2–3 m) is suspended from a fixed support. It carries a fixed graduated scale and below it a heavy fixed load. This load keeps the wire straight and free from kinks. The wire itself serves as a reference. The experimental wire B of almost equal length is also suspended from the same support close to the reference wire. A vernier scale is attached at the free end of the experimental wire. This vernier scale can slide against the main scale attached to the reference wire.
Surface Tension
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Surface tension is a property of liquid surfaces that causes them to behave as if covered by a stretched elastic membrane. Liquids tend to minimize their surface area due to this tension.
S = F/l, where S is surface tension, F is force, and l is length
- Surface tension is demonstrated by phenomena such as soap films forming on wire frames and the tendency of liquid drops to be spherical.
- The surface tension is the force acting per unit length across a line drawn on the liquid surface.
- Liquids tend to decrease their surface area.
- Examples: a soap film bounded by the ring and a thread loop. The loop takes a circular shape when the film inside the loop is pricked.
- Examples: the wire slides to the closing arm of the U-shaped frame so that the surface shrinks.
Pressure Difference Across a Curved Liquid Surface (Drop)
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The pressure inside a liquid drop is greater than the pressure outside due to surface tension. This difference in pressure depends on the surface tension (S) of the liquid and the radius (R) of the drop. The pressure on the concave side is always greater than the pressure on the convex side.
['F1 = 2πRS (Force due to surface tension)', 'F2 = P1πR^2 (Force due to external air pressure)', 'F3 = P2πR^2 (Force due to internal liquid pressure)', 'P - P = 2S/R (Pressure difference across a curved liquid surface)', 'ΔS cosθ (area of projection of area ΔS)']
- Consider a hemispherical section of a liquid drop.
- Forces acting on the hemisphere are due to: surface tension (F1), external air pressure (F2), and internal liquid pressure (F3).
- The force due to surface tension acts along the periphery of the hemisphere and is given by F1 = 2πRS, where S is the surface tension and R is the radius.
- The force due to external air pressure is calculated by considering the pressure (P1) acting on a small area (ΔS) of the hemispherical surface. The component of this force along the symmetry axis (OX) is P1ΔS cosθ, where θ is the angle between the radius through ΔS and OX.
- The resultant force due to the external air pressure is F2 = P1πR^2, where P1 is the pressure just outside the surface.
- The resultant force due to the internal liquid pressure is F3 = P2πR^2, where P2 is the pressure just inside the surface.
- For equilibrium of the hemispherical surface, F1 + F2 = F3.
- This leads to the equation P2 - P1 = 2S/R, which shows that the pressure inside the drop is greater than the pressure outside by 2S/R.
- For an air bubble inside a liquid, the pressure inside the air bubble (concave side) is greater than the pressure in the liquid (convex side) by an amount 2S/R.
Rise of Liquid in a Capillary Tube
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When a capillary tube is dipped into a liquid, the liquid either rises or is depressed depending on the contact angle. If the angle is less than 90 degrees, the liquid rises; if greater, it's depressed. The height of the rise/depression depends on surface tension, radius of the tube, density of the liquid, and contact angle.
F = 2πrS cosθ
- Capillary Action: The phenomenon of liquid rising or being depressed in a narrow tube (capillary tube) when dipped in a liquid.
- Contact Angle (θ): The angle formed at the point of contact between a liquid surface and a solid surface. If θ < 90°, the liquid rises. If θ > 90°, the liquid is depressed.
- Forces involved at the contact point:
- Adhesive Force (Fs): The force of attraction between molecules of different substances (e.g., liquid and solid).
- Cohesive Force (Fl): The force of attraction between molecules of the same substance (e.g., liquid molecules).
- Weight (W): The gravitational force acting on the liquid.
- Equilibrium of Liquid in Capillary: The liquid rises/falls until the forces are balanced. These forces include:
- Force due to surface tension.
- Force due to air pressure above the liquid.
- Force due to liquid pressure below.
- Weight of the liquid column.
Viscosity and Flow Through Tubes
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Viscosity is the measure of a fluid's resistance to flow. The velocity gradient describes how the velocity of the fluid changes with distance. Poiseuille's equation describes the flow rate of a viscous fluid through a cylindrical tube.
['F = -η A (dv/dz)', '[η] = ML^(-1)T^(-1)', 'V/t = (πPr^4) / (8ηl)']
- Velocity Gradient: The change in velocity (dv) with respect to the distance (dz) perpendicular to the flow, denoted as dv/dz.
- Viscous Force: The force between fluid layers due to viscosity is proportional to the area (A) of the layer and the velocity gradient (dv/dz). F = -ηA(dv/dz), where η is the coefficient of viscosity. The negative sign indicates the force opposes motion.
- Coefficient of Viscosity (η): A measure of a fluid's resistance to flow. Its SI unit is N-s/m2, and its CGS unit is poise (1 poise = 0.1 N-s/m2). The dimensions of η are ML-1T-1.
- Temperature Dependence: The coefficient of viscosity is highly dependent on temperature.
- Poiseuille's Equation: Describes the volume flow rate (V/t) of a viscous fluid through a cylindrical tube of radius r and length l, with a pressure difference P between the ends.
- Stokes' Law: Describes the viscous force F acting on a spherical body of radius r moving at a speed v through a fluid of viscosity η.
Terminal Velocity Experiment
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Describes an experiment to determine the viscosity of a liquid using a falling ball and measuring its terminal velocity.
η = [2(ρ − σ)gr^2] / [9(d/t)] where η = coefficient of viscosity, ρ = density of the solid, σ = density of the liquid, g = acceleration due to gravity, r = radius of the ball, d = length, t = time
The experiment involves dropping a metal ball through a viscous liquid and measuring the time it takes to travel between two points. The radius and mass of the ball are measured, and the terminal velocity is calculated. This data is used to determine the coefficient of viscosity of the liquid.
Critical Velocity and Reynolds Number
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Explains the concept of critical velocity, which is the maximum velocity at which a fluid flow remains steady. Introduces Reynolds number as a key factor in determining whether the flow is steady or turbulent.
N = (ρvD) / η where ρ = density of the fluid, v = velocity of the fluid, D = diameter of the tube, η = coefficient of viscosity. Flow is steady if N < 2000, turbulent if N > 3000, and unstable if 2000 < N < 3000.
Defines critical velocity as the largest velocity which allows a steady flow. Explains that the nature of flow (steady or turbulent) depends on the density, velocity, and coefficient of viscosity of the fluid, as well as the diameter of the tube. Reynolds number (N) is introduced as a dimensionless quantity used to predict the nature of flow.
Young's Modulus and Poisson Ratio
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Defines Young's modulus as the ratio of stress to strain and Poisson's ratio as the ratio of lateral strain to longitudinal strain. Presents example problems using these concepts.
Y = (T/A) / (l/L), Poisson's ratio = (∆d/d) / (l/L)
Young's modulus (Y) is defined as stress/strain = (T/A) / (l/L), where T is tension, A is area, l is elongation, and L is original length. Poisson's ratio is defined as (∆d/d) / (l/L), where ∆d is the change in diameter and d is the original diameter.
Young's Modulus and Elongation
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This section discusses Young's modulus, which relates stress and strain in a material, and how to calculate the elongation of a wire or rod under tension.
Y = (T/A) / (l/L) => l = (TL) / (AY)
- Young's Modulus (Y): The ratio of longitudinal stress to longitudinal strain. It's a measure of a material's stiffness.
- Longitudinal Stress: Force per unit area (T/A) acting on a material.
- Longitudinal Strain: Change in length divided by the original length (l/L).
- Formula for Elongation (l): l = (TL) / (AY), where T is tension, L is original length, A is cross-sectional area, and Y is Young's modulus.
Elastic Potential Energy
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Describes how to calculate the elastic potential energy stored in a wire when it's stretched.
U = 1/2 * (stress) * (strain) * (volume)
- Elastic Potential Energy (U): The energy stored in a material due to its deformation.
- U = 1/2 * (stress) * (strain) * (volume)
Excess Pressure Inside a Liquid Bubble
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Explains how pressure inside an air bubble in a liquid exceeds atmospheric pressure due to surface tension and depth.
Excess Pressure = (2 * Surface Tension) / Radius + (Density * g * Depth)
The pressure inside a bubble is greater than the atmospheric pressure due to two factors:
- Surface tension of the liquid.
- Hydrostatic pressure due to the depth of the bubble.
Viscosity
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Viscosity is the measure of a fluid's resistance to flow. It's like internal friction within the fluid.
['F = ηA dv/dx', 'Shearing stress = F/A = η dv/dx']
The magnitude of the force of viscosity is given by F = ηA dv/dx, where η is the coefficient of viscosity, A is the area, and dv/dx is the velocity gradient (rate of change of velocity with respect to distance). Shearing stress is defined as F/A = η dv/dx.
Terminal Velocity
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Terminal velocity is the constant speed that a freely falling object eventually reaches when the force of air resistance equals the force of gravity.
['6πηrv = (4/3)πr³ρg (at terminal velocity, neglecting buoyancy)', 'v = (2r²ρg) / (9η) (terminal velocity, neglecting buoyancy)']
When an object falls through a fluid (like air), it experiences a viscous force opposing its motion. At terminal velocity, the net force on the object is zero; thus, the weight of the object is balanced by the viscous force and the buoyant force (if significant). For a rain drop, the terminal velocity can be found by equating the weight of the drop (4/3 πr³ρg) to the viscous force (6πηrv), where r is the radius, ρ is the density of water, g is the acceleration due to gravity, η is the coefficient of viscosity and v is the terminal velocity.
Viscosity
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Viscosity is the property of a fluid (liquids and gases) that opposes the force which acts between two layers of a liquid.
F/A = -η(dv/dz)
Viscosity is a measure of a fluid's resistance to flow. It arises from the internal friction within the fluid as its layers move relative to each other. The viscous force is an electromagnetic force. The viscous force acting between two layers of a liquid is given by F/A = -η(dv/dz), where F is the viscous force, A is the area of the layer, η is the coefficient of viscosity, and dv/dz is the velocity gradient.
Terminal Velocity
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Terminal velocity is the constant speed that a freely falling object eventually reaches when the force of air resistance equals the force of gravity.
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When an object falls through a fluid, it experiences a drag force due to viscosity. As the object's speed increases, so does the drag force. Eventually, the drag force equals the weight of the object, and the net force becomes zero. At this point, the object stops accelerating and falls with a constant velocity called the terminal velocity.
Excess Pressure
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Excess pressure is the pressure difference between the inside and outside of a curved surface, such as a soap bubble or a liquid droplet.
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The excess pressure inside a soap bubble is related to the surface tension and the radius of the bubble. A smaller bubble will have a larger excess pressure than a larger bubble, given the same surface tension.
Capillary Action
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Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of, and in opposition to, external forces like gravity.
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Capillary action is caused by the interplay of cohesive forces (attraction between liquid molecules) and adhesive forces (attraction between liquid and solid molecules). The height to which a liquid rises in a capillary tube depends on the radius of the tube, the surface tension of the liquid, the contact angle between the liquid and the tube, and the density of the liquid.
Surface Properties
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Surface molecules experience different forces compared to molecules in the bulk of a liquid, leading to surface tension and other unique properties.
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Molecules at the surface of a liquid experience a net inward force because they are surrounded by fewer molecules than those in the bulk. This inward force creates surface tension, which makes the surface behave like a stretched membrane. This also influences properties such as contact angle.
Contact Angle
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The contact angle between a liquid and a solid is a measure of the wettability of the solid by the liquid. It is a property of the materials of the solid and the liquid.
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The contact angle is the angle formed at the point where a liquid-vapor interface meets a solid surface. It is determined by the balance of adhesive and cohesive forces. A contact angle less than 90 degrees indicates that the liquid wets the solid (adhesive forces are stronger), while a contact angle greater than 90 degrees indicates that the liquid does not wet the solid (cohesive forces are stronger).
Young's Modulus
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Young's modulus (Y) is a measure of a solid's stiffness or resistance to deformation under tensile or compressive stress. It relates stress (force per unit area) to strain (relative deformation).
Y = Stress / Strain = (F/A) / (ΔL/L₀), where F is the force applied, A is the cross-sectional area, ΔL is the change in length, and L₀ is the original length.
Young's modulus is defined as the ratio of tensile stress to tensile strain. It is a property of the material and indicates its stiffness. A higher Young's modulus implies that the material is more resistant to elastic deformation.
Poisson's Ratio
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Poisson's ratio (ν) describes the ratio of lateral strain (change in width) to axial strain (change in length) when a material is stretched or compressed. It indicates how much a material deforms in directions perpendicular to the applied force.
ν = - (Lateral Strain / Axial Strain) = - (Δd/d₀) / (ΔL/L₀), where Δd is the change in diameter/width, d₀ is the original diameter/width, ΔL is the change in length, and L₀ is the original length.
Poisson's ratio is a dimensionless quantity and is usually positive, although some materials have negative Poisson's ratios (auxetic materials). It provides information about the material's behavior under stress.
Bulk Modulus
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Bulk modulus (B) is a measure of a fluid's resistance to compression. It relates pressure change to volume change.
B = - (ΔP / (ΔV/V₀)), where ΔP is the change in pressure, ΔV is the change in volume, and V₀ is the original volume.
Bulk modulus is defined as the ratio of volumetric stress to volumetric strain. A higher bulk modulus indicates that the material is more resistant to compression.
Rigidity Modulus (Shear Modulus)
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Rigidity modulus (G), also known as shear modulus, measures a solid's resistance to deformation when subjected to shear stress (tangential force).
G = Shear Stress / Shear Strain = (F/A) / (Δx/L), where F is the tangential force, A is the area of the surface, Δx is the lateral displacement, and L is the height/thickness.
Rigidity modulus is defined as the ratio of shear stress to shear strain. It reflects how easily a material can be twisted or deformed by a tangential force.
Surface Tension
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Surface tension (γ) is the property of a liquid's surface that allows it to resist an external force. It's caused by cohesive forces between liquid molecules.
γ = F/L, where F is the force due to surface tension and L is the length along which the force acts.
Surface tension is defined as the force per unit length acting along the surface of a liquid. It is responsible for phenomena like capillary action and the formation of droplets.
Excess Pressure in Liquid Droplets and Bubbles
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Excess pressure refers to the pressure inside a liquid drop or bubble that's higher than the pressure outside due to surface tension.
For a liquid drop: ΔP = 2γ/r, where γ is surface tension and r is the radius. For a soap bubble: ΔP = 4γ/r
The excess pressure is caused by the surface tension trying to minimize the surface area. The formula depends on whether it's a liquid drop (one surface) or a soap bubble (two surfaces).
Capillary Action
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Capillary action is the ability of a liquid to flow in narrow spaces against the force of gravity. It's caused by surface tension and adhesive forces.
h = (2γ cos θ) / (ρgr), where h is the height of the liquid column, γ is the surface tension, θ is the contact angle, ρ is the density of the liquid, g is the acceleration due to gravity, and r is the radius of the capillary tube.
The height to which a liquid rises in a capillary tube depends on the surface tension, the contact angle between the liquid and the tube, the density of the liquid, and the radius of the tube.
Viscous Force and Terminal Velocity
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Viscous force is the frictional force exerted by a fluid on an object moving through it. Terminal velocity is the constant speed reached when the viscous force equals the gravitational force.
Viscous force (Stokes' Law): F = 6πηrv, where η is the viscosity of the fluid, r is the radius of the sphere, and v is the velocity. Terminal velocity: v_t = (2r²g(ρ_object - ρ_fluid)) / (9η)
The viscous force depends on the object's shape, size, and velocity, as well as the fluid's viscosity. Terminal velocity is reached when the net force on the object is zero.