Name: viscosity and how fluids flow
Uploaded: 2023-04-30T11:05:28.000Z
Duration: 1 min 17 s
Channel: JoVE

K12 Standards

ohios-learning-standards-and-model-curriculum

Ohio’s Learning Standards and Model Curriculum

9-12

Physics

Forces, Momentum And Motion

Air resistance and drag

Viscosity

水と蜂蜜の入った2つの瓶を考えてみましょう。落ちると水がこぼれるのが早くなりますが、蜂蜜はゆっくりと流れて表面に付着します。
観察された違いは、液体層間、または液体と周囲の材料との間の流体摩擦によるものです。この特性は流体粘度と呼ばれます。
流体の粘度は、流れが層流であるか乱流であるかに影響します。粘度が高いほど、流体がシート状に流れる可能性が高くなり、流れは層流になります。
粘度を測定する流体を、2つの平行なプレートの間に置いたとします。底板は静止していますが、天板は一定の速度で移動し、その下に流体の層を引きずります。
この引きず力の大きさは、層流速度、プレート面積、および粘度係数に正比例しますが、プレート間の距離に反比例します。したがって、粘度が高いほど、プレートを動かすために必要な力が大きくなります。

粘度

水をグラスに注ぐと、水は自由に素早く落ちますが、蜂蜜やメープルシロップをパンケーキに注ぐと、ゆっくりと流れて容器の表面にくっつきます。 異なる種類の液体の流れのこの違いは、液体層と液体および周囲の物質との間の流体摩擦によって生じます。 流体のこの性質は流体の粘度と呼ばれます。 この例では、水は蜂蜜やメープルシロップよりも粘度が低くなります。

粘度の SI 単位は、ニュートン秒/メートル 2 (N's/m2) またはパスカル秒 (Pa's) です。 粘度は温度に依存する量であり、液体ごとに数桁変化します。 たとえば、空気、水、蜂蜜の粘度は、20°C でそれぞれ 0.0181 mPa・s、1.002 mPa・s、2000・10000 mPa・s です。 水の粘度は温度の上昇とともに低下し、100°C で 0.282 mPa·s と計算されます。

流体の流れには、層流と乱流という 2 つのタイプがあります。 層流では、流体の層が混合せずに平行に流れます。 ただし、粘度により、層間だけでなく固定面との間でも抵抗が発生します。 上層の速度は下層よりも高くなります。 これにより、移動面から固定面に向かう方向に減少する速度勾配が作成されます。 乱流では、高速性と障害物により、渦や渦を介して層の混合が引き起こされます。 重大な速度は、流れ全体の方向以外の方向で発生します。

燃えている線香から出る煙は、流体の流れの良い例です。 お線香の先端では、煙がしばらく層流を示しながら滑らかに立ち上ります。 その後、加速する煙の速度は、煙と周囲の空気の間の抵抗によって渦巻き始める点に達し、乱流を示します。

このテキストは <a href="https://openstax.org/books/university-physics-volume-1/pages/14-7-viscosity-and-turbulence"> から編集されたものです。Openstax, University Physics Volume 1, Section 14.7: Viscosity and Turbulence.</a>

グラスに水を注ぐと、水は自由に素早く落ちますが、パンケーキの上に蜂蜜やメープルシロップを注ぐと、ゆっくりと流れてくっつきます...

Education

JoVE Core

物理学

Unit 1

ユニット1

Fluid Mechanics

流体力学

KEY TERMS AND CONCEPTS

主要用語と概念

viscosity and how fluids flow

Viscosity describes how easily a fluid flows. Water moves quickly, while honey and maple syrup flow slowly and stick to a surface. That difference comes from fluid friction between liquid layers and between a fluid and the material around it. In this example, water has a lower viscosity than honey and maple syrup.

Viscosity is measured in newton-second per meter squared (N·s/m2) or pascal-second (Pa·s). It also changes with temperature and can differ from one fluid to another by several orders of magnitude. At 20 °C, the viscosities of air, water, and honey are 0.0181 mPa·s, 1.002 mPa·s, and 2000–10000 mPa·s, respectively. For water, viscosity decreases as temperature increases, and it is calculated to be 0.282 mPa·s at 100 °C.

Fluid flow is often described as laminar flow or turbulent flow. In laminar flow, the fluid moves in parallel layers without mixing. Viscosity creates drag between these layers and with a fixed surface. The top layer moves faster than the lower layers, so the velocity changes from the moving surface toward the fixed surface.

In turbulent flow, high speed and obstacles cause the layers to mix through eddies and swirls. This means the fluid has significant motion in directions other than the main direction of flow. Smoke from a burning incense stick shows both patterns. It rises smoothly at first as laminar flow, then begins to swirl as drag with the surrounding air increases and turbulent flow starts.

Viscosity and How Fluids Flow

characteristics of fluids

Fluid Behavior Under Force

Fluid behavior under force differs from the behavior of solids. When a force acts parallel to the top surface of a solid, the solid resists it because of internal friction between its layers. This is called shearing resistance. If the force is removed, the shearing forces help the solid return to its original shape. Other deformation forces can also change the shape of a solid for a short time, as long as the force stays below a threshold magnitude. Solids usually keep their shape, which makes...

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density

Density, Floating, and Specific Gravity

Density helps explain why objects sink or float in a fluid. It is also useful for identifying a substance and for learning about its phase and substructure. The SI unit of density is kg/m3, and the cgs unit is g/cm3. Liquids and solids usually have densities that are roughly comparable. That matches the fact that their atoms are in close contact. Gases have much lower densities than liquids and solids because their atoms are separated by large amounts of empty space. Density does not always...

pressure of fluids

Fluid Pressure in Everyday Life

Fluid pressure appears in many everyday situations, including blood pressure and weather systems. The same force can have a very different effect depending on the area it acts on. A force spread over 1 mm² creates 100 times more pressure than the same force spread over 1 cm². That is why a sharp needle can pierce skin with a small force, while the same force from a finger usually does not. Pressure is especially important when studying fluids. A fluid does not offer much resistance to a force...

variation of atmospheric pressure

Atmospheric Pressure and Altitude

Atmospheric pressure and altitude are closely linked. As height increases, atmospheric pressure decreases. This happens because the gravitational force per unit area becomes smaller as we move away from Earth’s surface. To describe this change, the atmosphere is treated with a constant air temperature at a given altitude. The ideal gas law of thermodynamics is also used as a good approximation for the air. Let p(y) be the atmospheric pressure at height y. The density ρ at y, the temperature T...

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pascal's law

Pascal’s Principle in Static Fluids

Pascal’s principle explains how pressure changes move through a fluid that is not flowing. A fluid at rest is in static equilibrium. If the fluid is water, it is in hydrostatic equilibrium. In this state, the net force on any part of the fluid must be zero, or the fluid will begin to flow. Blaise Pascal described this idea in 1653 in Treatise on the Equilibrium of Liquids. He showed that when pressure changes in an enclosed fluid, that change is transmitted without loss through the fluid and...

application of pascal's law

Hydraulic Force Transfer with Pascal's Law

Hydraulic force transfer uses Pascal's law to move pressure through an enclosed fluid. Pascal showed experimentally that a change in pressure applied to a confined fluid is transmitted undiminished throughout the fluid and to the walls of its container. That idea is the basis of hydraulics, a major part of modern mechanical technology. Hydraulic systems appear in automotive brakes, hydraulic jacks, and many other machines. A simple hydraulic system uses two pistons at the same height. Because...

pressure gauges

Gauge vs Absolute Pressure

Pressure gauges often show pressure relative to atmospheric pressure. Many gauges, including those on scuba tanks, are set to read zero at atmospheric pressure. That reading is called gauge pressure. When the pressure inside the tank is higher than atmospheric pressure, the gauge shows a positive value. Some gauges can also measure negative pressure. This is useful in a vacuum chamber, which is a rigid chamber where some air has been pumped out. Because the pressure inside the chamber is below...

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archimedes' principle

Buoyant Force from Fluid Pressure

Archimedes' principle explains buoyant force in a fluid. An upward buoyant force acts on an object that is partially or fully immersed in a fluid. This force is equal to the weight of the fluid displaced by the object. To see why an object floats, imagine removing a submerged object from the fluid. The space it occupied would be filled by fluid with weight w_fl. That weight is supported by the surrounding fluid, so the buoyant force must match the weight of the displaced fluid. The principle...

density and archimedes' principle

How Buoyancy Depends on Density

Buoyancy and density explain why some objects float and others sink. A lump of clay sinks in water, but the same clay can float when it is shaped like a boat. The change in shape lets it displace more water, which increases the buoyant force even though its mass stays the same. The same idea applies to steel ships. An object with an average density lower than the surrounding fluid will float. The fluid has more mass in the same volume, so it has more weight. The buoyant force equals the weight...

accelerating fluids

Fluid Pressure in an Accelerating Elevator

Fluid pressure changes when a liquid accelerates inside a moving container. A beaker placed in an elevator that moves upward with constant acceleration is one example. In this setup, the pressure and buoyant force equations must be modified. To find the pressure difference, the liquid is modeled as a thin vertical cylinder with height h and an infinitesimal cross-sectional area, ΔS. Three vertical forces act on this small fluid element. The liquid below the bottom surface pushes upward, the...

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surface tension and surface energy

Why Liquids Form Drops and Float

Surface tension and surface energy explain why liquids behave the way they do at their surface. A paint brush moves freely in water, but its bristles stick together after it is taken out. That change is caused by surface tension. Inside a liquid, bulk molecules feel attractive forces from nearby molecules in all directions. At the surface, molecules feel a net downward pull from the molecules below them. Because of this imbalance, the liquid surface acts like a stretched membrane and tries to...

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excess pressure inside a drop and a bubble

excess-pressure-inside-a-drop-and-a-bubble

Surface Tension in Drops and Bubbles

Surface tension sets the pressure difference in small drops and bubbles. For a tiny liquid drop, gravity can be neglected, so the drop is treated as spherical. It can also be viewed as two equal hemispherical drops joined together by surface tension. The forces on the drop come from three sources. The liquid inside pushes outward, the air outside pushes inward, and surface tension acts along the diameter between the two hemispheres. Since surface tension is force per unit length, it creates a...

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buoyancy

Buoyant Force in Sinking and Floating

Buoyant force explains why objects float, sink, or stay suspended in a fluid. A fluid can be a liquid or a gas, and every object placed in it experiences this upward force. A metal ball sinks, while a rubber ball floats. A submarine also changes its buoyancy to move up or down. The force comes from pressure changes in the fluid. Fluid pressure increases with depth because the weight of the fluid above also increases. That means the bottom of an object feels more pressure than the top. This...

contact angle

Wetting and Surface Curvature

Wetting and surface curvature describe how a liquid behaves where it meets a solid. When a solid is dipped into a liquid, the liquid surface can bend near the contact line. In some cases, the liquid rises along the solid. In others, the surface becomes convex or depressed near the solid. This behavior comes from cohesive and adhesive forces. Adhesive force is the attraction between molecules of different materials, such as a solid and a liquid. Cohesive force is the attraction between...

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rise of liquid in a capillary tube

Capillary Action in Thin Tubes

Capillary action explains why liquid can rise or fall in a very thin tube compared with the liquid around it. A capillary is a narrow cylindrical tube, and the effect becomes noticeable when it is dipped into a liquid. The movement happens because two forces act in opposite directions. Cohesive forces make liquid molecules stick to each other, so the liquid tends to hold together and form a rounded shape. Adhesive forces pull the liquid toward the walls of the container. In a narrow glass...

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laminar and turbulent flow

Fluid Motion Patterns in Air and Water

Fluid dynamics studies fluids in motion. Scientists often use velocity vectors to show fluid motion in places like meteorology. For example, wind in the atmosphere can be mapped with vectors that show the speed and direction of the wind at each point. Another way to show fluid motion is with a streamline. A streamline shows the path of a small volume of fluid as it flows. When the flow changes with time, the streamlines do not match the flow lines. The fluid velocity is always tangential to...

equation of continuity

Fluid Flow in Narrowing Pipes

Fluid flow in a pipe is described by flow rate and average speed. Flow rate, or volume flow rate, is the amount of fluid that passes a point through an area in a given time. It depends on both how fast the fluid moves and the size and shape of the conduit. For a river or pipe, a greater speed usually means a greater flow rate. A larger cross-sectional area also increases flow rate. That is why the same fluid can move differently through conduits of different sizes. The relationship becomes...

bernoulli's equation

Fluid Pressure and Speed in Motion

Bernoulli's equation connects fluid pressure with fluid speed. It helps explain why moving fluids can create pressure changes in everyday situations. One example is when two trains pass each other quickly and seem to pull inward. The same effect can happen when two cars pass each other at high speed. As the air between the trains or cars speeds up, its pressure drops. The air on the outside stays at atmospheric pressure. That pressure difference creates a net inward force on the vehicles.

bernoulli's principle

Fluid Pressure and Speed Changes

Bernoulli's equation links fluid pressure and fluid speed in a static, incompressible fluid. For horizontal flow, where gravitational energy density stays constant, the equation is especially useful. In that setting, the result is known as Bernoulli's principle. Bernoulli's principle says that fluid pressure drops when speed increases, and pressure rises when speed decreases. This pressure-speed relationship helps explain how moving fluids behave in simple flow situations. It also provides a...

bernoulli's principle: applications

Bernoulli's Principle in Fluid Devices

Bernoulli's principle helps explain how many fluid devices work when the fluid moves at a constant height. It is useful for devices that use fast flow to create low pressure and for devices that measure fluid speed. Some devices use this pressure change to pull one fluid into another. These are called entrainment devices. A Bunsen burner uses an adjustable gas nozzle to entrain air for proper combustion. An atomizer uses a squeeze bulb to make a jet of air that pulls perfume drops into the...

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energy conservation and bernoulli's equation

energy-conservation-and-bernoulli-s-equation

Bernoulli’s Equation in Streamlined Flow

Bernoulli’s equation describes energy in a moving fluid. It comes from the conservation of energy principle, or the work-energy theorem, applied to an incompressible, inviscid fluid in laminar, steady, irrotational flow. The equation says that pressure energy, potential energy, and kinetic energy per unit volume stay constant along a streamline. Each term has the units of energy per unit volume. The kinetic energy per unit volume is called kinetic energy density, and the potential energy per...

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viscosity

Viscosity describes how easily a fluid flows. Water moves quickly, while honey and maple syrup flow slowly and stick to a surface. That difference comes from fluid friction between liquid layers and between a fluid and the material around it. In this example, water has a lower viscosity than honey and maple syrup. Viscosity is measured in newton-second per meter squared (N·s/m2) or pascal-second (Pa·s). It also changes with temperature and can differ from one fluid to another by several orders...

poiseuille's law and reynolds number

Fluid Resistance, Viscosity, and Flow Regime

Fluid resistance, viscosity, and flow regime shape how liquid moves through a horizontal tube. Fluid flows from high pressure to low pressure. The flow rate, Q, is the ratio of the pressure difference to the resistance in the tube. A larger pressure difference gives a higher flow rate. Poiseuille's law describes laminar flow of an incompressible fluid in a tube. In this relation, the factors that affect flow rate except pressure are grouped into resistance. Resistance depends on the tube size...

stokes' law

Terminal Velocity in a Viscous Fluid

Viscous forces act like friction in a liquid. They resist the relative motion of molecules and pull back on a solid body moving through the fluid. The size of this force depends on the body’s shape and size, its speed, and the liquid’s viscosity, density, and temperature. For a spherical solid object, the force in a fluid is described by Stokes’ law. This law applies only at low Reynolds number, when the flow is laminar and not turbulent. Experiments show that the force is proportional to the...

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key terms and concepts

Explore viscosity, fluid friction, and the difference between laminar and turbulent flow with clear high school examples like water, honey, and smoke.

Viscosity and How Fluids Flow | JoVE K12

physics

JoVE Core 物理学は、運動学、力学、流体力学、静力学、波動、音響などの基礎的な物理学の理論と応用を、簡潔でわかりやすいアニメーション ビデオで紹介します。さらに、科学者の実際のビデオでは、世界中の今日の研究所で行われている古典的および独自の研究実験における関連概念の適用を示しています。

Force

Forces affect every moment of our life. Our bodies are held to the Earth by force, and they are held together by the forces of charged particles. When we ...

Types of Forces

In most situations, forces can be grouped into two categories: contact forces and field forces.&#160; Contact forces occur as a result of direct physical ...

Newton's First Law: Introduction

Motion draws our attention. Motion itself can be beautiful, causing us to marvel at the forces needed to create spectacular sights, such as that of a ...

Newton's First Law: Application

Experience suggests that an object at rest remains at rest if left alone, and that an object in motion tends to slow down and stop unless some effort is ...

Internal and External Forces

Newton&#39;s first law states that a net external force causes a change in motion. External forces act on an object or system, originating outside of the ...

Newton's Second Law

Newton&#39;s second law is closely related to his first law of motion. It mathematically gives the cause-and-effect relationship between force and changes ...

Mass and Weight

Mass and weight are often used interchangeably in everyday conversation. For example,&#160; medical records often show our weight in kilograms, but never ...

Newton's Third Law: Introduction

Whenever one body exerts a force on a second body, the first body experiences a force equal in magnitude and opposite in direction, to the force that it ...

Newton's Third Law: Examples

Newton&#39;s third law states that every action has an equal and opposite reaction. Consider a swimmer pushing off the side of a pool. They push against ...

Second Law: Motion under Same Force

Newton&#39;s laws can be applied to bodies at rest and bodies in motion. Newton&#39;s first law is applied to bodies in equilibrium, whereas the second ...

second-law-motion-under-same-acceleration

Second Law: Motion under Same Acceleration

Newton&#39;s second law of motion applies to bodies moving under the same acceleration. For example, when a baggage tractor pulls luggage carts, each cart ...

Frictional Force

When a body is in motion, it encounters resistance because the body interacts with its surroundings. This resistance is known as friction, a common yet ...

Static and Kinetic Frictional Force

One of the simpler characteristics of sliding friction is that it is parallel to the contact surfaces between systems, and is always in a direction that ...

Frequency of Spring-Mass System

One interesting characteristic of the simple harmonic motion (SHM) of an object attached to a spring is that the angular frequency, and the period and ...

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Newton's laws applied to complex problems

Content elaboration: Newton's laws of motion, especially the third law, can be used to solve complex problems that involve systems of many objects that move together as one (e.g., an Atwood machine). The equation $a=F_{net}/m$ that was introduced in physical science can be used to solve more complex problems involving systems of objects and situations involving forces that must themselves be quantified (e.g., gravitational forces, elastic forces, friction forces).

newton-s-laws-applied-to-complex-problems

Newton's laws applied to complex problems

Gravitation

In the years before Newton, a general belief prevailed that different laws governed objects in the sky than objects on Earth. When Kepler wrote down the ...

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Newton's Law of Gravitation

Our everyday observation tells us that all objects close to the Earth naturally tend to fall to the ground. Early philosophers assumed that this downward ...

gravitation-between-spherically-symmetric-masses

Gravitation Between Spherically Symmetric Masses

The gravitational potential energy between two spherically symmetric bodies can be calculated from the masses and the distance between the bodies, ...

/files/media/science-education/science-education-thumbs/14717.jpg

Gravity between Spherical Bodies

Newton&#39;s law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, ...

Apparent Weight

True weight is the measure of the gravitational force acting on an object. However, if the object accelerates, its measured weight is different from its ...

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Apparent Weight and the Earth's Rotation

Since all objects on the Earth&#39;s surface move through a circle every 24 hours, there must be a net centripetal force on each object, directed towards ...

the-principle-of-superposition-and-the-gravitational-field

The Principle of Superposition and the Gravitational Field

The principle of superposition applies to gravitational forces of objects that are sufficiently far apart. It states that the net gravitational force on a ...

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Gravitational force and fields

Content elaboration: Gravitational interactions are very weak compared to other interactions and are difficult to observe unless one of the objects is extremely massive (e.g., the sun, planets, moons). The force law for gravitational interaction states that the strength of the gravitational force is proportional to the product of the two masses and inversely proportional to the square of the distance between the centers of the masses, $F_{g}=(G\cdot m_{1}\cdot m_{2})/r^{2}$. The proportionality constant, G, is called the universal gravitational constant and has a value of $6.674\cdot10^{-11}m^{3}/(kg\cdot s^{2})$. Problem solving may involve calculating the net force for an object between two massive objects (e.g., Earth-moon system, planet-sun system) or calculating the position of such an object given the net force. The strength of an object's (i.e., the source's) gravitational field at a certain location, g, is given by the gravitational force per unit of mass experienced by another object placed at that location, $g=F_{g}/m$. Comparing this equation to Newton's second law can be used to explain why all objects on Earth's surface accelerate at the same rate in the absence of air resistance. While the gravitational force from another object can be used to determine the field strength at a particular location, the field of the object is always there, even if the object is not interacting with anything else. The field direction is toward the center of the source. Given the gravitational field strength at a certain location, the gravitational force between the source of that field and any object at that location can be calculated. Greater gravitational field strengths result in larger gravitational forces on masses placed in the field. Gravitational fields can be represented by field diagrams obtained by plotting field arrows at a series of locations. Field line diagrams are excluded from this course. Distinctions between gravitational and inertial masses are excluded. A scale indicates weight by measuring the normal force between the object and the surface supporting it. The reading on the scale accurately measures the weight if the system is not accelerating. However, if the scale is used in an accelerating system, as in an elevator, the reading on the scale does not equal the actual weight. The scale reading can be referred to as the "apparent weight." This apparent weight in accelerating elevators can be explained and calculated using force diagrams and Newton's laws.

Gravitational force and fields

Simple Harmonic Motion

Simple harmonic motion is the name given to oscillatory motion for a system where the net force can be described by Hooke&#39;s law. If the net force can ...

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characteristics-of-simple-harmonic-motion

Characteristics of Simple Harmonic Motion

The key characteristic of the simple harmonic motion is that the acceleration of the system and, therefore, the net force are proportional to the ...

/files/media/science-education/science-education-thumbs/12763.jpg

oscillations-about-an-equilibrium-position

Oscillations about an Equilibrium Position

Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable ...

Hooke's Law

Hooke&#39;s law states that stress applied to a material is directly proportional to the strain it experiences within the elastic limit. For ductile ...

Stress

When a force is applied on a body, it undergoes deformation. In order to restore the body to its original shape and/or size, an opposite or restoring ...

Strain and Elastic Modulus

The quantity that describes the deformation of a body under stress is known as strain. Strain is given as a fractional change in either length, volume, or ...

Elasticity

Elasticity is the ability of an object to withstand the effects of distortion and to return to its original size and shape once the forces causing ...

Elastic forces

Content elaboration: Elastic materials stretch or compress in proportion to the load they support. The mathematical model for the force that a linearly elastic object exerts on another object is $F_{elastic}=k\Delta x$ where $\Delta x$ is the displacement of the object from its relaxed position. The direction of the elastic force is always toward the relaxed position of the elastic object. The constant of proportionality, k, is the same for compression and extension and depends on the "stiffness" of the elastic object.

Elastic forces

Friction force (static and kinetic)

Content elaboration: The amount of kinetic friction between two objects depends on the electric forces between the atoms of the two surfaces sliding past each other. It also depends upon the magnitude of the normal force that pushes the two surfaces together. This can be represented mathematically as $F_{k}=\mu_{k}F_{N}$, where $\mu_{k}$ is the coefficient of kinetic friction that depends upon the materials of which the two surfaces are made. Sometimes friction forces can prevent objects from sliding past each other, even when an external force is applied parallel to the two surfaces that are in contact. This is called static friction, which is mathematically represented by $F_{s}\le\mu_{s}F_{N}$ The maximum amount of static friction possible depends on the types of materials that make up the two surfaces and the magnitude of the normal force pushing the objects together, $F_{smax}=\mu_{s}F_{N}$ As long as the external net force is less than or equal to the maximum force of static friction, the objects will not move relative to one another. In this case, the actual static friction force acting on the object will be equal to the net external force acting on the object, but in the opposite direction. If the external net force exceeds the maximum static friction force for the object, the objects will move relative to each other and the friction between them will no longer be static friction, but will be kinetic friction.

Friction force (static and kinetic)

Drag Force and Terminal Speed

An interesting force in everyday life is the force of drag on an object when it is moving in a fluid. Like friction, the drag force always opposes the ...

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Density

Density is an important characteristic of substances, crucial in determining whether an object sinks or floats in a fluid. Its SI unit is kg/m3, and its ...

When water is poured into a glass, it falls freely and quickly, whereas if honey or maple syrup is poured over a pancake, it flows slowly and sticks to ...

Air resistance and drag

Content elaboration: Liquids have more drag than gases. When an object pushes on the particles in a fluid, the fluid particles can push back on the object according to Newton's third law and cause a change in motion of the object. This is how helicopters experience lift and how swimmers propel themselves forward. Forces from fluids are quantified using Newton's second law and force diagrams. Factors that affect air resistance and drag and the determination of terminal velocity may be included.

Introduction to Vectors

To define some physical quantities, there is a need to specify both magnitude as well as direction. For example, when the U.S. Coast Guard dispatches a ...

Vector Algebra: Graphical Method

Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the ...

Forces in two dimensions
• Adding vector forces
• Motion down inclines
• Centripetal forces and circular motion

Content elaboration: Net forces will be calculated for force vectors with directions between $0^{\circ}$ and 360° or a certain angle from a reference (e.g., 37° above the horizontal). Vector addition can be done with trigonometry or by drawing scaled diagrams. Problems can be solved for objects sliding down inclines. The net force, final velocity, time, displacement and acceleration can be calculated. Inclines will either be frictionless or the force of friction will already be quantified. Calculations of friction forces down inclines from the coefficients of friction and the normal force will not be addressed in this course. An object moves at constant speed in a circular path when there is a constant net force that is always directed at right angles to the direction of motion toward the center of the circle. In this case, the net force causes an acceleration that shows up as a change in direction. If the force is removed, the object will continue in a straight-line path. The nearly circular orbits of planets and satellites result from the force of gravity. Centripetal acceleration is directed toward the center of the circle and can be calculated by the equation $ac=v^{2}/r$, where v is the speed of the object and r is the radius of the circle. This expression for acceleration can be substituted into Newton's second law to calculate the centripetal force. Since the centripetal force is a net force, it can be equated to friction (unbanked curves), gravity, elastic force, etc., to perform more complex calculations.

Forces in two dimensions

Viscosity and How Fluids Flow

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