Name: spherical coordinates for 3d positioning
Uploaded: 2023-05-16T08:18:40.000Z
Duration: 1 min 23 s
Channel: JoVE

K12 Standards

Tennessee Academic Standards

High School

Physics

motion-and-stability-forces-and-interactions

Motion and Stability: Forces and Interactions

PHYS.PS2.4

Spherical Coordinates

Spherical coordinates, an extension of polar coordinates, describe a vector&#39;s position in three-dimensional space.
Unlike cylindrical coordinates, which describe systems with cylindrical symmetry, spherical coordinates are applied to explain systems with spherical symmetry.
A vector in a spherical coordinate system is defined using the radial, polar, and azimuthal scalar components.
The radial component, which ranges from zero to infinity, specifies the vector&#39;s distance from its origin.
The polar angle ranges from zero to &#960; and measures the angle between the positive z-axis and the vector.
The azimuthal angle, which ranges from zero to two &#960;, measures the angle between the x-axis and the orthogonal projection of the vector onto the xy-plane.
A surface with a constant radius traces a sphere in a three-dimensional spherical coordinate system. On the other hand, surfaces with a constant polar angle form half-cones, and those with a constant azimuthal angle form half-planes.
The transformation equations are used to convert a vector in spherical coordinates to cartesian coordinates. Similarly, conversion from spherical to cylindrical coordinates is possible.

spherical coordinates

Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere&#39;s radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and gravitational fields. One of the other common applications of spherical coordinates is in the Earth&#39;s latitude and longitude system, which is used for navigational purposes.
Spherical coordinates belong to the family of curvilinear coordinates. These are the extension of polar coordinates and are used to describe a vector&#39;s position in three-dimensional space. A vector in a spherical coordinate system is defined using the radial, polar, and azimuthal scalar components. The radial component, which ranges from zero to infinity, specifies the vector&#39;s distance from its origin. The polar angle ranges from zero to &#960; and measures the angle between the positive z-axis and the vector. The azimuthal angle, which ranges from zero to 2&#960;, measures the angle between the x-axis and the orthogonal projection of the vector onto the xy-plane. A surface with a constant radius traces a sphere in a three-dimensional spherical coordinate system. On the other hand, a surface with a constant polar angle forms a half-cone, and a surface with a constant azimuthal angle forms a half-plane.
The transformation equations are used to convert a vector in spherical coordinates to Cartesian coordinates and cylindrical coordinates.

Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe ...

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Education

JoVE Core

Unit 1

Vectors and Scalars

KEY TERMS AND CONCEPTS

spherical coordinates for 3d positioning

Spherical coordinates describe position in three-dimensional space using radius, polar angle, and azimuthal angle. They are especially useful for systems with spherical symmetry, where the shape matches a sphere.

Compared with Cartesian coordinates, spherical coordinates can make calculations simpler. To describe a sphere, Cartesian coordinates need three values. Spherical coordinates can use just one parameter, the radius. That is why they are common in science and engineering problems such as electric fields and gravitational fields.

Spherical coordinates are also used in Earth’s latitude and longitude system for navigation. They belong to the family of curvilinear coordinates, which extend polar coordinates into three dimensions. In this system, the radial component gives the distance from the origin and ranges from zero to infinity. The polar angle ranges from zero to pi and measures the angle between the positive z-axis and the vector.

The azimuthal angle ranges from zero to 2 pi. It measures the angle between the x-axis and the projection of the vector onto the xy-plane. A constant radius makes a sphere. A constant polar angle forms a half-cone, and a constant azimuthal angle forms a half-plane. The transformation equations convert spherical coordinates into Cartesian coordinates and cylindrical coordinates.

Spherical Coordinates for 3D Positioning

introduction to scalars

Scalars in Everyday Physics

Scalar quantities are physical quantities that can be described completely by one number and a unit. A class period can last 50 min, a gas tank can hold 65 L, and the distance between two posts can be 100 m. In each case, the value is fully given with no direction needed. The word scalar is a synonym for number. Time, mass, distance, length, volume, temperature, and energy are all examples of scalar quantities. When two scalar quantities have the same units, they can be added or subtracted...

introduction to vectors

Vectors, Scalars, and Direction

Vectors and scalars describe physical quantities in different ways. Some quantities need both magnitude and direction to be fully specified. In those cases, the quantity is called a vector quantity. Examples include displacement, velocity, position, force, and torque. A quantity with only magnitude and no direction is a scalar quantity. The difference matters in everyday examples. If a rescue team knows only the distance to a distress signal, that is not enough to reach it quickly. They also...

vector components in the cartesian coordinate system

vector-components-in-the-cartesian-coordinate-system

Breaking Vectors into x and y Parts

Vectors in a Cartesian coordinate system are often split into x and y parts. This makes it easier to describe direction and size on a rectangular plane. It also matches the way people give directions in everyday life, such as east, west, north, south, and an angle of travel. In a plane, a point is written as an ordered pair, (x, y). A vector is described in a similar way with its vector coordinates. The x-coordinate of the vector is its x-component, and the y-coordinate is its y-component.

polar and cylindrical coordinates

Using Polar and Cylindrical Coordinates

Polar and cylindrical coordinates are useful ways to describe position when rotation or cylindrical shape is involved. Cartesian coordinates work well for displacement, velocity, and forces, but they can become awkward when an object turns or curves around an axis. In the polar coordinate system, a point in a plane is described by two values. The first is the radial coordinate, or the distance from the origin. The second is the angle made by the radial vector, usually measured from the...

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Spherical coordinates describe position in three-dimensional space using radius, polar angle, and azimuthal angle. They are especially useful for systems with spherical symmetry, where the shape matches a sphere. Compared with Cartesian coordinates, spherical coordinates can make calculations simpler. To describe a sphere, Cartesian coordinates need three values. Spherical coordinates can use just one parameter, the radius. That is why they are common in science and engineering problems such...

vector algebra: graphical method

Adding and Subtracting Vectors Graphically

Vectors can be added, subtracted, and multiplied by scalars. The vector sum of two or more vectors is called the resultant vector, or resultant. Geometric rules help find the resultant and its direction in a plane. For two vectors, the parallelogram rule shows how to build the sum. First, move one vector in parallel so both tails start at the same point. Then draw lines through the heads that are parallel to the opposite vector. The diagonal from the shared origin gives the resultant vector. <br...

vector algebra: method of components

Vector Components for Adding Vectors

Vector components make it easier to add, subtract, and multiply vectors. The parallelogram rule and graphical methods can become hard to use when the math gets complex. One option is to draw vectors to scale, as in navigation, and estimate their lengths and directions from the graph. Another option is the method of components. In many situations, the size and direction of each vector are already known, and the goal is to find the resultant, or combined vector, of several vectors. A real-world...

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scalar product (dot product)

Dot Product and Work in Physics

The dot product links two vectors to a single number. That number is called a scalar. In physics, this idea helps describe work and energy. A common example is work done by a force on an object. Force is a vector, and displacement is also a vector. The work is defined as the scalar product of the force vector and the displacement vector. To find the dot product, multiply the magnitudes of the two vectors by the cosine of the angle between them. The angle can be measured from either vector.

vector product (cross product)

Right-Hand Rule for Vector Multiplication

Vector multiplication gives a vector product, also called the cross product. Its magnitude equals the product of the two vectors times the sine of the angle between them. The result points perpendicular to both vectors. The direction of this perpendicular vector is found with the right-hand thumb rule. Because a plane has two possible perpendicular directions, the rule shows which one to choose. To use it, rotate the first vector about the perpendicular line until it lines up with the second...

scalar and vector triple products

Triple Products in Vectors and Volume

Scalar and vector triple products extend the rules for multiplying two vectors to three vectors. A scalar product gives a scalar, while a vector product gives a vector. These same ideas help define two useful triple-product forms. The scalar triple product is the dot product of one vector with the cross product of two others. It produces a scalar quantity. When the vectors are rotated in a cyclic order, the result stays the same, so this form is associative. It also shows the projection of one...

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gradient and del operator

Gradient Direction on Scalar Fields

The gradient helps describe how a scalar field changes in space. It gives both the size and the direction of the greatest rate of change. In physics and mathematics, this makes it a key tool for studying functions and fields. A simple mountain example shows why direction matters. A slope does not mean much unless it is measured in a specific direction. That is why the directional derivative is used. It is a vector that gives the slope and the direction of change. The gradient of a scalar...

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divergence and curl

Vector Fields: Sources, Sinks, and Rotation

Vector fields can spread outward, pull inward, or rotate around a point. Divergence describes the net outward flow of flux from a small volume through a closed surface as the volume gets very small. In simple terms, it shows how much a vector field spreads out from a point. When the flux moves outward, the divergence is positive. A point with positive divergence is often called a source. When the field contracts toward a point, the divergence is negative. That point is often called a sink.

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second derivatives and laplace operator

Using Del Operators to Build Laplacians

The del operator can form important second-order expressions from gradients, divergence, and curl. These combinations matter in mathematics and physics. They help describe how scalar and vector functions change at a deeper level. For a scalar function, the curl of its gradient is one such expression. For a vector function, the divergence of a curl is another. The curl of a gradient and the divergence of a curl are always zero. The divergence of the gradient of a scalar function leads to the...

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line, surface, and volume integrals

Work, Circulation, and Flux Integrals

Work, circulation, flux, and volume integrals describe how scalar and vector fields add up over a path, surface, or region. For high school science, these ideas help connect math to motion, flow, and stored mass or energy. A line integral for a vector field adds the dot product of a vector function and a tiny displacement vector along a chosen path. If the path is closed, the result is a closed-line integral. The value around a closed path is called the circulation of the vector field. A...

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divergence and stokes' theorems

Divergence and Stokes Theorems in Vector Fields

Divergence and Stokes theorems help connect vector fields to the shapes they pass through. A vector field gives a value with direction at each point, such as electric flow or wind flow. These theorems extend Green's theorem to higher dimensions and also generalize the fundamental theorem of calculus. The divergence theorem focuses on the dot product of a vector field. It lets a difficult surface integral become a volume integral, and it can also work in reverse. With this theorem, the rate of...

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key terms and concepts

Learn how spherical coordinates describe 3D position with radius, polar angle, and azimuthal angle, plus real uses in fields and navigation.

Spherical Coordinates for 3D Positioning | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

Texas Essential Knowledge and Skills

PHYS.5

PHYS.5.B

Kinematic Equations - II

The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant ...

Kinematic Equations - III

The first two kinematic equations have time as a variable, but the third kinematic equation is independent of time. This equation expresses final velocity ...

Kinematic Equations: Problem Solving

When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the ...

Acceleration Vectors

In everyday conversation, accelerating means speeding up. Acceleration is a vector in the same direction as the change in velocity, &#916;v, therefore the ...

Solving Problems in Physics

Problem-solving is the ability to apply general physical principles to specific situations, usually expressed by equations. It is an essential skill in ...

Algebraically solve problems including constant velocity and constant acceleration in onedimension. Algebraically solve problems including constant velocity and constant acceleration in onedimension. EXPLANATION:  Students can use the models they have developed in PHYS.PS2.1 to evaluate systems. For a system to undergo constant acceleration, the net force on the object must be constant throughout the problem. Algebraic problem solving should be extended to include proportional reasoning, beyond simple manipulation of variables.

PHYS.PS2.2

Angular Velocity and Displacement

Uniform circular motion is motion in a circle at a constant speed. Although this is the simplest case of rotational motion, it is very useful for many ...

Angular Velocity and Acceleration

We previously discussed angular velocity for uniform circular motion, however not all motion is uniform. Envision an ice skater spinning with their arms ...

rotation-with-constant-angular-acceleration-i

Rotation with Constant Angular Acceleration - I

If angular acceleration is constant, then we can simplify equations of rotational kinematics, similar to the equations of linear kinematics. This ...

rotation-with-constant-angular-acceleration-ii

Rotation with Constant Angular Acceleration - II

Kinematics is the description of motion. The kinematics of rotational motion discusses the relationships between rotation angle, angular velocity, angular ...

Equation of Rotational Dynamics

Angular variables are introduced in rotational dynamics. Comparing the definitions of angular variables with the definitions of linear kinematic ...

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Relating Angular And Linear Quantities - I

If the rotational definitions are compared with the definitions of linear kinematic variables from motion along a straight line and motion in two and ...

relating-angular-and-linear-quantities-ii

Relating Angular And Linear Quantities - II

In the case of circular motion, the linear tangential speed of a particle at a radius from the axis of rotation is related to the angular velocity by the ...

Moment of Inertia

The comparability between linear and angular velocities, linear and angular accelerations, and the kinematic equations of translational and rotational ...

moment-of-inertia-and-rotational-kinetic-energy

Moment of Inertia and Rotational Kinetic Energy

The rotational kinetic energy of a body is equal to half the square of its angular speed and the moment of inertia.
This relationship between the ...

Algebraically solve problems involving arc length, angular velocity, and angular acceleration. Algebraically solve problems involving arc length, angular velocity, and angular acceleration. EXPLANATION: Though not explicitly stated, it is beneficial to develop this standard in the same manner which PHYS.PS2.1 is used to develop PHYS.PS2.2. In doing so, students can parallel rotational properties to translational properties, e.g., arc length can be seen as the rotational equivalent to displacement in the translational world. In doing so, radians becomes a logical unit of measure for rotational displacement. Since neither torque, nor moment of inertia are addressed in this course, discussions can be limited to considering only kinematics and not venturing into the realm of rotational dynamics.

PHYS.PS2.3

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 ...

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 ...

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 ...

Drawing Free-body Diagrams: Rules

The first step in describing and analyzing most phenomena in physics involves the careful drawing of a free-body diagram. Free-body diagrams are useful in ...

Free Body Diagrams: Examples

Solving problems that involve forces is easy using free-body diagrams. A free-body diagram is a sketch showing all the external forces that are acting on ...

Vector Components in the Cartesian Coordinate System

Vectors are usually described in terms of their components in a coordinate system. Even in everyday life, we naturally invoke the concept of orthogonal ...

Polar and Cylindrical Coordinates

The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on ...

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 ...

Vector Algebra: Method of Components

It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like ...

Work Done Over an Inclined Plane

The center-of-mass framework helps to easily describe the work done on rigid bodies. Since the internal forces in a rigid body do no work, they can be ...

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Projectile Motion

An object thrown in the air follows a parabolic path under the influence of Earth&#39;s gravitational force. The motion of such an object is called ...

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Projectile Motion: Equations

Projectile motion is commonly observed in our day-to-day life. For example, a basketball thrown by a player, an arrow shot from a bow, and kids jumping ...

Projectile Motion: Example

The theory of projectile motion is very useful for players of several sports to improve their performance. For example, a javelin thrower needs to throw ...

Use free-body diagrams to illustrate the contact and noncontact forces acting on an object. Use the diagrams in combination with graphic or component based analysis with Newton’s first and second laws to predict position of the object on which the forces act in a constant net force scenario. Use free-body diagrams to illustrate the contact and noncontact forces acting on an object. Use the diagrams in combination with graphic or component based analysis with Newton’s first and second laws to predict position of the object on which the forces act in a constant net force scenario. EXPLANATION: The concept of net force and Newton’s laws have been introduced in 8.PS2.3 and 8.PS2.4. At that time, free-body diagrams are introduced as a tool to represent the forces acting on an object. In further developing the use of free-body diagrams, it is now appropriate to include vectors that must be evaluated to parallel and perpendicular components. This includes objects on inclined planes as well as projectile motion as addressed in PHYS.PS2.13.

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 ...

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 ...

Free-falling Bodies: Introduction

All objects, neglecting air resistance, fall with the same acceleration towards the Earth&#39;s center due to the force exerted by the Earth&#39;s ...

Free-falling Bodies: Example

An object falling without any air resistance under the influence of gravitational force is said to be in free-fall. For free-falling bodies, the ...

Acceleration due to Gravity on Earth

According to Newton&#39;s law of gravitation, the gravitational force on a body is proportional to its mass. According to Newton&#39;s second law of ...

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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Hooke's Law and Simple Harmonic Motion

Source: Ketron Mitchell-Wynne, PhD,&#160;Asantha Cooray, PhD, Department of Physics &#38; Astronomy, School of Physical Sciences, University of ...

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 ...

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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Elastic Potential Energy

Elastic potential energy is the energy stored as a result of the deformation of an elastic object, such as the stretching of a spring. An object is ...

Gather evidence to defend the claim of Newton’s first law of motion by explaining the effect that balanced forces have upon objects that are stationary or moving at a constant velocity.Gather evidence to defend the claim of Newton’s first law of motion by explaining the effect that balanced forces have upon objects that are stationary or moving at a constant velocity. EXPLANATION: Students should be able to discuss mass as a measurement of the amount of inertia in an object, with a unit of kg. Beginning as early as 5.PS2.3 students have been developing a understanding of gravity. Experiments determining the rate of acceleration of objects in freefall can be used to determine earth’s gravitational field strength which can then be used to develop an object’s gravitational mass. A discussion of Hook’s Law and subsequent use of a spring with a known spring constant can be used to determine an object’s inertial mass.

PHYS.PS2.5

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 ...

Linear Momentum

The term momentum is used in various ways in everyday language, most of which are consistent with the precise scientific definition. Generally, momentum ...

Force and Momentum

Force and momentum are intimately related. Force acting over time can change momentum, and Newton&#39;s second law of motion can be stated in its most ...

Impulse

According to Newton&#8217;s second law of motion, the rate of change of the momentum of an object is the net external force acting on it. The total change ...

Impulse-Momentum Theorem

The total change in the motion of an object is proportional to the total force vector acting on it and the time over which it acts. This product is called ...

Conservation of Momentum: Introduction

The total momentum of a system consisting of N interacting objects is constant in time or is conserved. A system must meet two requirements for its ...

Conservation of Momentum: Problem Solving

Solving problems using the conservation of momentum requires four basic steps:

	Identify a closed system, where the total mass is constant, and no net ...

Types Of Collisions - I

When two objects come in direct contact with each other, it is called a collision. During a collision, two or more objects exert forces on each other in a ...

Using experimental evidence and investigations, determine that Newton’s second law of motion defines force as a change in momentum, F = ∆p/∆t. Using experimental evidence and investigations, determine that Newton’s second law of motion defines force as a change in momentum, F = ∆p/∆t. EXPLANATION:  Previous examinations of Newton’s second law have been limited to instances with constant forces. This standard expands that discussion to include instances where the objects interact with each other. To maximize the quality of experimental investigations, magnets might be used to create situations where objects “collide” in an elastic manner. Newton’s second law can be expressed as F = ma. Viewing acceleration as a change in velocity over a period of time, one arrives at F=m(∆v/∆t). Distributing mass into this equation yields F = (mV-mVo)/∆t. A final recognition that momentum (p) is a property described by an object’s mass and velocity allows for substitution to produce F = ∆p/∆t.

PHYS.PS2.6

Spherical Coordinates for 3D Positioning

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