Two Dimensional Motion also called Planar Motion is any motion in which the objects being analyzed stay in a single plane. When analyzing such motion, we must first decide the type of coordinate system we wish to use. The most common options in engineering are rectangular coordinate systems, normal-tangential coordinate systems, and polar coordinate systems. Any planar motion can potentially be described with any of the three systems, though each choice has potential advantages and disadvantages.
The normal-tangential coordinate system centers on the body in motion. The origin point will be the body itself, meaning that the position of the particle in the n-t coordinate system is always "zero".
The tangential direction t-direction is defined as the direction of travel at that moment in time the direction of the current velocity vectorwith the normal direction n-direction being 90 degrees counterclockwise from the t-direction. The diagram below shows a particle following a curved path with the current normal and tangential directions.
Two Dimensional Kinematics in Normal-Tangential Coordinate Systems
Normal-tangential coordinate systems work best when we are observing motion from the perspective of the body in motion, such as being a passenger in a car or plane. In such cases, we would define ourself as the origin point and "forward" would be the tangential direction. An important distinction between the rectangular coordinate system and the normal-tangential coordinate system is that the axes are not fixed in the normal-tangential coordinate system.
If we go back to the car example, the" forward" or tangential direction will turn with the car, but the "east" direction or the x-direction will remain constant no matter which way the car is pointed. The the way the coordinate system is defined, the position of the particle is always set to be at the origin point.
The velocity is also always set to be in the tangential direction, and thus there is no velocity in the n-direction. The variable "v"is the body's current speed. To find the acceleration, we need to take the derivative of the velocity function. This may seem simple, but there is a new thing to consider in that the u t unit vector is not constant. The means a change in speed can cause an acceleration or a change in direction which would change the ut direction can cause an acceleration.
Going back to our car example this makes some intuitive sense. We can feel accelerations, and we would be able to feel acceleration if we suddenly stepped on the gas and increased our speed, but we would also be able to feel the acceleration if we took a tight turn at a constant speed.
Going back to the derivative, we will use the product rule, taking the derivative of one piece at a time. V dot is the rate of change of speed of the body, which is called the tangential acceleration. Going back to our car analogy this is the acceleration we would experience from pressing the gas or brake pedals.
The other piece of our derivative is the speed times the derivative of a unit vector, which we will need to analyze further. When thinking about the derivative of a rotating unit vector, we think about rotating the coordinate system by a small amount d theta.The rectangular coordinate system or Cartesian plane provides a means of mapping points to ordered pairs and ordered pairs to points. This is called a one-to-one mapping from points in the plane to ordered pairs.
The polar coordinate system provides an alternative method of mapping points to ordered pairs. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates. This correspondence is the basis of the polar coordinate system. Note that every point in the Cartesian plane has two values hence the term ordered pair associated with it. The first coordinate is called the radial coordinate and the second coordinate is called the angular coordinate.
Every point in the plane can be represented in this form. These formulas can be used to convert from rectangular to polar or from polar to rectangular coordinates. Direct application of the second equation leads to division by zero. The polar representation of a point is not unique. Also, the value of r can be negative. Every point in the plane has an infinite number of representations in polar coordinates. However, each point in the plane has only one representation in the rectangular coordinate system.
Note that the polar representation of a point in the plane also has a visual interpretation. Positive angles are measured in a counterclockwise direction and negative angles are measured in a clockwise direction. The line segment starting from the center of the graph going to the right called the positive x-axis in the Cartesian system is the polar axis.
The line segments emanating from the pole correspond to fixed angles. To plot a point in the polar coordinate system, start with the angle. If the angle is positive, then measure the angle from the polar axis in a counterclockwise direction. If it is negative, then measure it clockwise. If the value of r is positive, move that distance along the terminal ray of the angle. If it is negative, move along the ray that is opposite the terminal ray of the given angle.
Now that we know how to plot points in the polar coordinate system, we can discuss how to plot curves. The general idea behind graphing a function in polar coordinates is the same as graphing a function in rectangular coordinates.Mechanics - 220.127.116.11 - Position and Unit Vectors in Polar Coordinates
This process generates a list of ordered pairs, which can be plotted in the polar coordinate system. Finally, connect the points, and take advantage of any patterns that may appear. The function may be periodic, for example, which indicates that only a limited number of values for the independent variable are needed.
Identify the curve and rewrite the equation in rectangular coordinates.The Mechanics Map is an open textbook for engineering statics and dynamics containing written explanations, video lectures, worked examples, and homework problems. All content is licensed under a creative commons share-alike license, so feel free to use, share, or remix the content.
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Newtonian Mechanics Video Introduction 1. Statically Equivalent Systems Video Introduction 4. Engineering Structures Video Introduction 5.
Friction and Friction Applications Video Introduction 6. Tipping 6. Particle Kinematics Video Introduction 7. Work and Energy in Particles Video Introduction 9. Impulse Momentum Methods Video Introduction Work and Energy Methods Video Introduction Welcome to the Mechanics Map Digital Textbook: The Mechanics Map is an open textbook for engineering statics and dynamics containing written explanations, video lectures, worked examples, and homework problems.
Mechanics Basics: 1. Rigid Body Kinematics: Newton's Second Law for Rigid Bodies: Vector and Matrix Math: A1. Moment Integrals: A2. Video Introduction.Two Dimensional Motion also called Planar Motion is any motion in which the objects being analyzed stay in a single plane.
When analyzing such motion, we must first decide the type of coordinate system we wish to use. The most common options in engineering are rectangular coordinate systems, normal-tangential coordinate systems, and polar coordinate systems.
Any planar motion can potentially be described with any of the three systems, though each choice has potential advantages and disadvantages. The polar coordinate system uses a distance r and an angle theta to locate a particle in space.
The origin point will be a fixed point in space, but the r-axis of the coordinate system will rotate so that it is always pointed towards the body in the system. The variable "r" is also used to indicate the distance from the origin point to the particle. The theta axis will then be 90 degrees counter clockwise from the r-axis with the variable "theta" being used to show the angle between the r-axis and some fixed axis that does not rotate. The diagram below shows a particle with a polar coordinate system.
Polar coordinate systems work best in systems where a body is being tracked via a distance and an angle such as a radar system tracking a plane. In cases such as this, the raw data from this in the form of an angle and a distance would be direct measures of theta and r respectively.
Polar coordinate systems will also serve as the base for extended body motion, where motors and actuators can directly control things like r and theta. The way the coordinate system is defined, the r-axis will always point from the origin point to the body. The distance from the origin to the point is defined as "r" with no component of the position being in the theta direction. To find the velocity, we need to take the derivative of the position function over time. Since the distance r can change over time as well as the direction r changing over time to track the body, we need to worry about the derivative of r as well as the derivative of the unit vector.
Like we did with the normal-tangential systems, we will use the product rule and then substitute in a value for the derivative of the unit vector. To find the acceleration, we need to take the derivative of the velocity function. As all of these terms, including the unit vectors, change over time, we will need to use the product rule extensively.
The u-r term will split into two terms, and the u-theta term will split into three terms. Again we will need to substitute in values for the derivatives of the unit vectors similar to before, but it is worth mentioning that the derivative of the u-theta vector as it rotates counter clockwise is in the negative u-r direction. After substituting in the derivatives of the unit vectors and simplifying the function we arrive at our final equation for the acceleration.
Though this final equation has a number of terms, it is still just two components in vector form. Just as with the normal-tangential coordinate system, we will need to remember that we will need to split the single vector equation into two separate scalar equations.
In this case we will have the equation for the terms in the r direction and the equation for the terms in the theta direction.
A radar tracking station gives the following raw data to a user at a given point in time. Based on this data, what is the current velocity and acceleration in the r and theta directions? What is the current velocity and acceleration in the x and y directions? A spotlight is tracking an actor as he moves across the stage. If the actor is moving with a constant velocity as shown below, what values do we need for the spotlight angular velocity theta dot and spotlight angular acceleration theta double dot so that the spotlight remains fixed on the actor?
In the polar coordinate system the r direction always points from the origin point to the body.
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The variable "r" is also used to indicate the distance between the two points. The theta direction will always be 90 degrees counter-clockwise from the r direction.
Theta is also used to indicate the angle between the r direction and some fixed axis used for reference. The u-r and u-theta vectors represent unit vectors in the r and theta directions respectively. The derivatives of the u-r and u-theta unit vectors.
Notice that the derivative of the u-theta vector is in the negative u-r direction.In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. However, there are other ways of writing a coordinate pair and other types of grid systems.
The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
The reference point analogous to the origin of a Cartesian system is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth. Polar Graph Paper: A polar grid with several angles labeled in degrees.
Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics. In mathematical literature, the polar axis is often drawn horizontal and pointing to the right. This point is plotted on the grid in Figure.
Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction.
11.3: Polar Coordinates
Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry. When given a set of polar coordinates, we may need to convert them to rectangular coordinates. Trigonometry Right Triangle: A right triangle with rectangular Cartesian coordinates and equivalent polar coordinates. To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.
Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated below.
This corresponds to the non-uniqueness of polar coordinates. Multiple sets of polar coordinates can have the same location as our first solution.
Previously, we learned how a parabola is defined by the focus a fixed point and the directrix a fixed line. Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph. Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates. Polar equations can be used to generate unique graphs.
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Modify, remix, and reuse just remember to cite OCW as the source. Course Home Syllabus. Review: Vectors. Week 1: Kinematics. Week 2: Newton's Laws. Week 3: Circular Motion. Week 5: Momentum and Impulse.
Week 6: Continuous Mass Transfer. Week 7: Kinetic Energy and Work. Week 8: Potential Energy and Energy Conservation. Week 9: Collision Theory. Week Rotational Motion.Ike, C. Journal of Computational Applied Mechanics50 1 Charles Ike.
Journal of Computational Applied Mechanics50, 1, Journal of Computational Applied Mechanics; 50 1 : Toggle navigation. Abstract In this work, the Mellin transform method was used to obtain solutions for the stress field components in two dimensional 2D elasticity problems in terms of plane polar coordinates. The Mellin transform was similarly used to obtain the Mellin transformed stress field components. The use of Mellin transform inversion formula yielded the solutions to the 2D elasticity problem in the physical space domain variables.
Specific illustration was considered of the solution by using the Mellin transform method for the Flamant problem and the Mellin transform solutions found to agree with solutions from the literature.
Fundamentals of Engineering Elasticity Second Edition. World Scientific Publishing Co. London, Theory of Elasticity ct Direct Methods. June Introduction to plane problems subject Application of plane stress, plane strain and revolution symmetric axisymmetric models. Mathematical Theory of Elasticity Second Edition. Atkin and N.
An introduction of the theory of elasticity. Longman Group Ltd, London Sitharam and L. Govinda Reju. Hamidzadeh and R.