Vector Basics. Lecture 1 Vector Basics

Transcription

1 Lecture 1 Vector Basics Vector Basics We will be using vectors a lot in this course. Remember that vectors have both magnitude and direction e.g. a, You should know how to find the components of a vector from its magnitude and direction a a cos a asin You should know how to find a vector s magnitude and direction from its components a a a tan 1 a / a Was of writing vector notation F ma F ma F ma a a a Projection of a Vector and Vector Components When we want a component of a vector along a particular direction, it is useful to think of it as a projection. The projection alwas has length a cos, where a is the length of the vector and is the angle between the vector and the direction along which ou want the component. You should know how to write a vector in unit vector notation ˆ ˆ a a i a j or a ( a, a ) acos sin a cos Projection of a Vector in Three Dimensions n vector in three dimensions can be projected onto the - plane. The vector projection then makes an angle from the ais. Now project the vector onto the ais, along the direction of the earlier projection. The original vector a makes an angle from the ais. a

2 Vector Basics Seeing in 3 Dimensions You should know how to generalie the case of a -d vector to three dimensions, e.g. 1 magnitude and directions a,, Conversion to,, components a asin cos a asin sin a a cos Conversion from,, components a a a a cos tan 1 1 Unit vector notation: a a a a kˆ a / a a / a a 7. Which of these show the proper projection of the vector onto the ais?. I. B. II. C. III. D. IV. E. None of the above. I. II. III. IV. a a a a Note bout Right-Hand Coordinate Sstems three-dimensional coordinate sstem MUST obe the right-hand rule. Curl the fingers of our RIGHT HND so the go from to. Your thumb will point in the direction. Right Handed Coordinate Sstems 8. Which of these coordinate sstems obe the right-hand rule?. I and II. B. II and III. C. I, II, and III. D. I and IV. E. IV onl. I. II. III. IV.

3 Vector Math Projection of a Vector: Dot Product Vector Inverse Just switch direction Vector ddition Use head-tail method, or parallelogram method Vector Subtraction Use inverse, then add Vector Multiplication Two kinds! Scalar, or dot product Vector, or cross product B B B B B B B B Vector ddition b Components ( B )ˆ i ( B ) ( B ) kˆ The dot product sas something about how parallel two vectors are. The dot product (scalar product) of two vectors can be thought of as the projection of one onto the direction of the other. B B cos cos Components B B B B ( cos )B B ( B cos ) Projection of a Vector: Dot Product The dot product sas something about how parallel two vectors are. The dot product (scalar product) of two vectors can be thought of as the projection of one onto the direction of the other. B B cos cos Components B B B B B Projection is ero Derivation How do we show that B B B B? Start with ˆ j kˆ B B B ˆ j Bkˆ Then B ( ˆ ˆ ˆ ˆ i j kˆ) ( Bi B j Bkˆ) ( B B B kˆ) ( B B B kˆ) kˆ ( B B ˆ j Bkˆ) But 0; kˆ 0; kˆ 0 1; 1; kˆ kˆ 1 So B B ˆ j B ˆ j kˆ Bkˆ B B B

4 The cross product of two vectors sas something about how perpendicular the are. You will find it in the contet of rotation, or twist. B Bsin Cross Product Recall angular momentum L r p Torque r F B B sin Direction perpendicular to both and B (right-hand rule) B B Components (mess) kˆ B sin B B B ( B B )ˆ i ( B B ) ( B B ) kˆ Derivation How do we show th at B ( B B )ˆ i ( B B ) ( B B ) kˆ? Start with ˆ j kˆ B B B B kˆ Then But So ( B B B ( kˆ) ( B B B kˆ) B kˆ) ( B B kˆ; kˆ ˆ; j kˆ 0; 0; kˆ kˆ 0 B B B kˆ B B kˆ kˆ B kˆ B B kˆ) kˆ ( B B B kˆ) Gradients and Gravit Vector Fields Height contours h, are proportional to potential energ U = mgh. If ou move along a contour, our height does not change, so our potential energ does not change. If ou move downhill, on sa a 6% grade, it means the slope is 6/100 (for ever 100 m of horiontal motion, ou move downward b 6 m) Grade and gradient mean the same thing. 6% grade is a gradient of lim h / dh / d F du / dl d mgh / dl mg dh / dl F = mg sin dl mg dh vector field is one where a quantit in space is represented b both magnitude and direction, i.e b vectors. The vector field bears a close relationship to the contours (lines of constant potential energ). The steeper the gradient, the larger the vectors. The vectors point along the direction of steepest descent, which is also perpendicular to the lines of constant potential energ. Imagine rain on the mountain. The vectors are also streamlines. Water running down the mountain will follow these streamlines. Side View

5 Surface vs. Volume Vector Fields Vector Field Due to Gravit In the eample of the mountain, note that these force vectors are onl correct when the object is ON the surface. The actual force field anwhere other than the surface is everwhere downward (toward the center of the Earth. The surface creates a normal force everwhere normal (perpendicular) to the surface. The vector sum of these two forces is what we are showing on the contour plot. Side View When ou consider the force of Earth s gravit in space, it points everwhere in the direction of the center of the Earth. But remember that the strength is: GMm F rˆ r This is an eample of an inverse-square force (proportional to the inverse square of the distance). Notice that the actual amount of force depends on the mass, m: GMm F rˆ r It is convenient to ask what is the force per unit mass. The idea is to imagine putting a unit test mass near the Earth, and observe the effect on it: F GM rˆ g( r) rˆ m r g(r) is the gravitational field. Idea of Test Mass Meaning of g(r) F GM 9. What are the units of rˆ g( r) rˆ?. Newtons/meter (N/m) B. Meters per second squared (m/s ) C. Newtons/kilogram (N/kg) D. Both B and C E. Furlongs/fortnight m r

6 Meaning of g(r) 10. What is another name for g(r) in F GM rˆ g( r) rˆ? m r. Gravitational constant B. Gravitational energ C. cceleration of gravit D. Gravitational potential E. Force of gravit Gravitational Field We can therefore think of the action-at-a-distance of gravit as a field that permeates all of space. We draw field lines that show both the direction and strength of the field (from the densit of field lines). The field cannot be seen or touched, and has no effect until ou consider a second mass. What happens if we have two equal masses? Superposition just vector sum the two fields. Gravitational Field of Two Equal Masses gain, think of adding a small test mass. The force vectors show the direction and strength of the force on such a test mass. We can draw field lines that follow the force vectors. We will be using these same concepts when we talk about electric charge in Lecture, and the electric field in Lecture 3.