Motion in Two and Three Dimensions

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This article is a topic within the subject Higher Physics 1A.

Contents

Introduction

When examining multiple dimensions, directions are very important. It is important to always include the direction when giving vector quantities as answers.

Vectors

[1]

  • Scalar quantities have magnitude, but no direction (e.g. temperature).
  • Vector quantities have both direction and magnitude (e.g. acceleration).

Notation

[2]

  • a - Most textbooks and typed media display vector quantities as bold
  • ḇ - In handwriting, vectors are written as a regular character with a tilde (~) underneath.
  • Sometimes, vectors are denoted by a letter with a small arrow above it.

Drawing, Adding and Subtracting Vectors

[3] Vectors are drawn as arrows pointing in the direction of the vector.

  • The length of the vector represents its magnitude.
  • The negative of a vector is the exact same vector, drawn in the opposite direction.

PHYS1131 vectors 1.jpg

Vectors are added by drawing them head to tail (but their directions are not changed).

  • The resultant (or sum) vector is then simply the vector from the tail of the first vector to the head of the last.
  • Vector addition is commutative and so can be done in any order.[4]
  • Vectors are subtracted by drawing them in the opposite direction. Subtraction may be thought of as "adding negative vectors".
  • Vectors may be multiplied with a scalar by multiplying the vector's magnitude with the scalar.

PHYS1131 Vectors 2.jpg

If these vector diagrams are drawn to scale, they can be used to directly measure the resultant vector. However, this method is only practical for two-dimensional problems, and is somewhat inaccurate[5].

Vector Components and Unit Vectors

Any vector may be broken into components which lie along the coordinate axes. Usually, the orthogonal x,y,z axes are used, but this may vary depending on the question. Breaking a vector into its components requires simple trigonometry.[6]

PHYS1131 Vectors 3.jpg

A vector may be expressed either as its magnitude and direction, or as its various components. This is best done using unit vectors. A unit vector is defined as a single unit in a specified direction. They are essentially a shorthand way of writing the directions for vector components[7]. Unit vectors are written with a circumflex (^) above them, and also use bold font in typed media.

  • î represents one unit in the x direction.
  • ĵ represents one unit in the y direction.
  • represents one unit in the z direction.

For example:

  • u = 3î + 4ĵ - 7
  • The vector u is 3 units from the origin in the x direction, 4 units in the y direction, and 7 units in the negative z direction.

Using Components for Vector Calculations

Vector components can be used for accurate vector calculations.[8]

  • Vectors may be split into components using trigonometry
  • Vectors may be added by adding all x components together, adding all y components together, etc
    • Eg. If u = 3î + 4ĵ and v = -2î + ĵ,
u+v = 3î + 4ĵ -2î + ĵ
u+v = (3-2)î + (4+1)ĵ
u+v = î + 5ĵ
  • Vector subtraction works the same way as addition.
    • Eg. If u = 3î + 4ĵ and v = -2î + ĵ,
u-v = 3î + 4ĵ -(-2î + ĵ)
u-v = (3+2)î + (4-1)ĵ
u-v = 5î + 3ĵ
  • Scalar multiplication may be done by multiplying each component by the scalar.
    • Eg. If u = 4î + 5ĵ,
4u= 4(4)î + 4(5)ĵ
4u= 16î + 20ĵ
  • Vector components may be converted back to magnitude and direction using Pythagoras and trigonometry.
    • The angle, or direction, is best worked out by drawing vector diagrams and using trigonometry (not by measuring the vector diagrams).
    • The magnitude of is the length of the vector, which is found from the components using Pythagoras. Pythagoras works the same in three dimensions as for two.
For a vector u = xî + yĵ + z:
Magnitude of u = u = sqrt(x2+y2+z2)


Uniform Circular Motion

If an object moves in a circular path with constant speed then it is undergoing uniform circular motion. The following statements apply to objects undergoing uniform circular motion[9]:

  • Tangential velocity. The object's velocity is always tangential to the circle. The magnitude of this velocity (the speed) is constant.
  • Centripetal acceleration. The object is always accelerating radially inwards (perpendicular to the tangential velocity). It is this acceleration that keeps it in a circular path. The acceleration has constant magnitude, but its direction changes so that it is always perpendicular to the velocity.
    • Since the acceleration is perpendicular to the velocity, it will not affect the magnitude of the velocity (the speed). This is why the speed remains constant.

PHYS1131 circular motion 1.jpg

Circular Motion Formulas

When dealing with circular motion, it is often helpful to use angular motion. Angular motion is measured anticlockwise and uses radians (not degrees).[10]

  • Angular displacement, θ, is measured in radians.
    • θ = s/r. Angular displacement is displacement divided by the radius of the circle.
  • Angular velocity, ω, is measured in radians per second.
    • ω = v/r = dθ/dt. Angular displacement is velocity divided by radius. It is also the derivative of angular displacement with respect to time.
  • Angular acceleration, α, is measured in radians per second squared.
    • α = a/r = dω/dt. Angular displacement is acceleration divided by radius. It is also the derivative of angular velocity with respect to time.

Uniform Circular Motion Formulas

  • ac = v2/r

The centripetal acceleration (ac) is equal to velocity squared, divided by the radius of the circle. It is directed towards the center of the circle[11].

  • T = (2πr)/v

The period (T) is the time taken to complete one full revolution. It is equal to 2π times the radius, all divided by the velocity[12].


Non-uniform circular motion

An object undergoing non-uniform circular motion will have tangential acceleration as well as radial (centripetal) acceleration. The speed will not be constant.[13]

PHYS1131 circular motion 3.jpg

Non-uniform circular motion formulas

  • at = dv/dt

Tangential acceleration (at) is calculated y differentiating the tangential velocity, and is directed parallel to the tangential velocity. The tangential acceleration will be positive if the object is speeding up, or negative if slowing down[14].

  • atotal = ar + at

Total acceleration (atotal) is composed of both the tangential and the radial acceleration. To calculate the total acceleration, use vector addition to combine the two acceleration components[15].


Relative Velocity

Frames of Reference

A frame of reference is a set of axes or coordinates which are used to make measurements of objects inside the frame[16]. Frames of reference are nothing new - it is just a name used to describe where you are measuring from. The observer's frame of reference is at rest relative to the observer[17].

Relative velocity and acceleration

Velocity and acceleration will appear different depending on the frame of reference of the observer[18]. For example, imagine a beetle walking radially outwards on a steadily spinning plate. An observer standing on the plate (same frame of reference as the beetle) will see the beetle move in a straight line (1). However, an observer using a stationary frame of reference (standing beside the spinning plate) will see the beetle move in a spiral path (2).

PHYS1131 relative motion 1.jpg

Calculations with relative velocity and acceleration

When examining the motion of an object in another frame of reference, the motion of that frame of reference relative to the observer must also be taken into account[19].

Imagine two frames of reference. Let the first frame A be at rest. The second frame (B) is moving with constant velocity (v') A.

  • Time t=0 is defined as the time when both frames of reference are at the same point in space. Note that t=0 can be redefined if necessary, but this will need to be taken into account in your calculations.
  • rA = rB + v't
If an object is at position rB with respect to observer B, then its position rA with respect to observer A must also take into account the motion of the x',y',z' frame (v't).
  • vA = vB + v'
An object's velocity relative to A is the sum of the object's velocity relative to B and B's velocity relative to A.
  • aA = aB
Since B is not accelerating relative to A, the acceleration of an object with respect to B is the same as that with respect to A.

Now imagine that B is accelerating with respect to A. The equations have been modified to take into account this acceleration (a').

  • rA = rB + v't + a't2
  • vA = vB + v' + a't
  • aA = aB + a'


Projectile Motion

Projectile motion is the analysis of objects projected into the air or space. This describes any object that has been thrown, dropped, shot from a gun, etc. In this course, two assumptions are made about projectiles[20]:

  1. They experience a constant gravitational acceleration downwards.
  2. Air resistance has negligible effect on the motion.

These two assumptions make projectile analysis very simple, since the only force acting is gravity, which causes a known downwards acceleration. The projectile's motion can thus be split into two components:

  • Vertical motion with constant acceleration
  • Horizontal acceleration with constant velocity (zero acceleration)

Analysis of these components is straightforward because vertical acceleration is known (gravity), and initial velocity components can be calculated using vector components. The horizontal and vertical components are independent of each other.[21].


End

This is the end of this topic. Click here to go back to the main subject page for Higher Physics 1A.

References

Textbook refers to Serway & Jewett, Physics for Scientists and Engineers (Brooks/Cole , 8th ed, 2010)
(Slides) refers to those distributed by Wolfe, J (2012) on his First Year Physics site

  1. Textbook, pp. 57-63.
  2. Kinematics notes, p5 (slides).
  3. Textbook, pp. 57-63
  4. Textbook, p. 58
  5. Textbook, pp. 60-61
  6. Textbook, pp.60-61
  7. Textbook, p. 62
  8. Textbook, pp.61-62
  9. Textbook, pp.86-87
  10. Textbook, pp. 278-279
  11. Textbook, p87
  12. Textbook, p87
  13. Textbook, pp. 88-89
  14. Textbook, p88
  15. Textbook, p88
  16. Textbook, pp.105-106
  17. Textbook, p. 90
  18. Textbook, p. 90
  19. Textbook, pp. 90-91
  20. Textbook, p. 79
  21. Textbook, p. 80
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