01 lect(week 2), Lecture notes of Accelerator Physics

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Trigonometry

Trigonometry

h

h

o sin   h h a cos   a o

h

h

tan  

Trigonometry



h

h

(^1) o  sin         h h (^1) a  cos

 a o

h

h

1  tan

Trigonometry

 a o

h

h

1  tan 

14. 0 m

2. 25 m

tan

1

 

1.5 Scalars and Vectors By convention, the length of a vector arrow is proportional to the magnitude of the vector. 8 N 4 N

Arrows are used to represent vectors. The

direction of the arrow gives the direction of

the vector.

Vectors—Figures 1.9–1.

• (^) Vectors show magnitude and displacement, drawn as a ray.

Vector addition III—Figure 1.

• (^) Refer to Example 1.5.(Skier goes 1km N and 2 km E What is R?)

1 2 2 2

d   

tan 64. 3

1 2 O

 

• Vector additional II—Figure 1.
• Components of vectors II—Figure 1.

Finding components—Figure 1.

• (^) Refer to worked Example 1.6. D =3 and    = o E =4.5 m and  = o D x = D Cos (-45) = +2. D y = D Sin (-45) = -2. E x = E Sin (37) = +2. E y = ُE Cos )37) = +3. Sin (-45)=Sin (315) = (-1) x Sin(45)

The three finalist in a contest are brought to the center of a large flat field. Each is given a meter stick, a compass, a calculator, and a shovel ( in a different order for each contestant ) the following : three displacement ; 72.4m, 32 degrees east of north ; 57.3m, 36 degrees south of west 17.8m, straight south The three displacement lead to the point where the keys to a new Porsche are buried. Two contestants start measuring immediately, but the winner first calculates where to go. What does she ? calculate

Calculations using components II—Figure 1.

• (^) See worked examples 1.7 (three displacements) and

1.8. A =72.4 and the angle =58o

B =57.3 and the angle = o C =17.8 and the angle = o R x = -7. R y = 9.  =129 west of east^ = 39 o West of North

Unit vectors—Figures 1.23–1.

• (^) Assume vectors of magnitude 1 with no units exist in each of the three standard dimensions.
• (^) The x direction is termed i , the y direction is termed j , and the z direction, k.
• (^) A vector is subsequently described by a scalar times each component. A = A x i + A y j + A z k
• (^) Refer to Example 1.9.

A = A

x

î +A

y

j B = B

x

î + B

y j **or Find R = A

• B

1. R** = (A x î +A y j) + (B x î +B y j) 2) R = (A x

• B x )i + (A y +B y )j 3) = R x î + R y j Example The sum of the following three vectors A = 2i +3j , B = -i +2j and C = 4i -j is equal to, choose ? the correct answer a)7i + 6j b)5i + 4j c)3i + 2j