Idea Transcript
Physics 40 Vectors
Cartesian Coordinate System • Also called rectangular coordinate system • x- and y- axes intersect at the origin • Points are labeled (x,y)
Polar Coordinate System – Origin and reference line are noted – Point is distance r from the origin in the direction of angle θ, ccw from reference line – Points are labeled (r,θ)
Cartesian to Polar Coordinates • r is the hypotenuse and θ an angle
y tan θ = x = r
x2 + y2
θ must be ccw from positive x axis for these equations to be valid
Example • The Cartesian coordinates of a point in the xy plane are (x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point. Solution: r=
x2 + y2 =
( −3.50 m) 2 + ( −2.50 m) 2 = 4.30 m
−2.50 m tan ( = ) 35.5° −3.50 m −1
= θ 180° + 35.5= ° 216°
Polar to Cartesian Coordinates • Based on forming a right triangle from r and θ • x = r cos θ • y = r sin θ
Vector quantities have both magnitude and direction The length of the vector represents the magnitude of the vector. The orientation represents the direction or angle of the vector.
A
Example: velocity of 2 km/hr, 30 degrees north of east. 2 km/hr is the magnitude, 30 degrees north or east, the direction. Scalars only have magnitude: T = 82 degrees Celsius.
Writing Vectors • Text books use BOLD. Hard to write bold! • The vector is written with an arrow over head and includes both magnitude and direction.
• Magnitude (length of vector) is written with no arrow or as an absolute value:
A= A= magnitude= 2km
2km 30
A= ( 2km, 30 NE )
A= ( A,θ )
Vector Example • A particle travels from A to B along the path shown by the dotted red line – This is the distance traveled and is a scalar
• The displacement is the solid line from A to B – The displacement is independent of the path taken between the two points – Displacement is a vector
Equality of Two Vectors • Two vectors are equal if they have the same magnitude and the same direction • A = B if A = B and they point along parallel lines • All of the vectors shown are equal
Unit Vectors • A unit vector is a dimensionless vector with a magnitude of exactly 1. • Unit vectors are used to specify a direction and have no other physical significance • The symbols ˆi ,ˆj, and kˆ represent unit vectors • They form a set of mutually perpendicular vectors (called a basis) • Any vector with component Ax and Ay can be written in terms of unit vectors:
A = Ax ˆi + Ay ˆj + Az kˆ
Right Handed Coordinate System
iˆ × ˆj = kˆ
Vectors in 3-D The orientation of ijk doesn’t matter as long as they form a right-handed system!
r
WARNING! Vector Algebra is Weird!! Vectors have a different rules of operation for addition, subtraction, multiplication and division than ordinary real numbers!!!! Since vectors have magnitude and direction you can not always just simply add them!!!!
A & B co-linear
A & B NOT colinear
Adding Vectors The Graphical Method •Draw vectors to scale. •Draw the vectors “Tip to Tail.” •IMPORTANT: the angle of a vector is relative its own tail! •The resultant, R, is drawn from the tail of the first to the head of the last vector. •Use a ruler to MEASURE the resultant length. •Use a protractor to MEASURE the resultant angle.
Adding Vectors The Graphical Method • When you have many vectors, just keep repeating the process until all are included • The resultant is still drawn from the origin of the first vector to the end of the last vector
Adding Vectors Head to Tail
R=A + B = B + A When two vectors are added, the sum is independent of the order of the addition. – This is the commutative law of addition – A+B=B+A – The sum forms the diagonal of a Parallelogram!
Adding Vectors: Graphical Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find graphically (a) A + B, (Report all angles counterclockwise from the positive x axis. Use a scale of:
1 unit = 0.5 m
5.2m, 60
Subtraction of Vectors A = C + (− B )
Subtracting Vectors • If A – B, then use A+(-B) • The negative of the vector will have the same magnitude, but point in the opposite direction • Two ways to draw subtraction:
How do you go from A to -B ?
The distance between two vectors is equal to the magnitude of the difference between them! More on this later…
R= A + B
1. 2.
B 3.
R
Draw 1st vector tail A at origin Draw 2nd vector B with tail at the head of the 1st vector, A. The angle of B is measured relative to an imaginary axis attached to the tail of B. The resultant is drawn from the tail of the first vector to the head of the last vector.
A
R= A − B A
B R
R
OR B
A
Subtracting Vectors: Very Important for 2-D Kinematics (Chapter 4)
∆r v= ∆t
∆v a= ∆t
Parallelogram Method Addition & Subtraction
R= X + Y
R= X − Y
Vector Question When three vectors, A, B, and C are placed head to tail, the vector sum is:
A + B+ C = 0
If the vectors all have the same magnitude, the angle between the directions of any two adjacent vectors is a. 30 b. 60 c. 90 d. 120 e. 150
Adding Vectors: Graphical Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find graphically (a) A + B, (b) A − B, (c) B − A Report all angles counterclockwise from the positive x axis. Use a scale of:
1 unit = 0.5 m
5.2m, 60
3m,330
3m,150
Vector Addition Components Method
C= A + B
Unit Vectors • A unit vector is a dimensionless vector with a magnitude of exactly 1. • Unit vectors are used to specify a direction and have no other physical significance • The symbols ˆi ,ˆj, and kˆ represent unit vectors • They form a set of mutually perpendicular vectors (called a basis) • Any vector with component Ax and Ay can be written in terms of unit vectors:
A = Ax ˆi + Ay ˆj + Az kˆ
Vector Components A x = A cos θ A A y = A sin θ A
= A
Ax + Ay 2
2
Ay θ A = tan Ax -1
A= ( A x , A y )
A = ( A,θ A )
Vector Addition Components Method
C= A + B
Vector Addition Components Method
C= A + B
C y =A y +By
C x =A x +Bx
Vector Addition Components Method = C
Cx + C y 2
2
Cy θ C = tan Cx -1
C x =A x +Bx
C= A + B
C y =A y +By
Adding Vectors Using Unit Vectors • R=A+B
(
) (
R = Ax ˆi + Ay ˆj + Bx ˆi + By ˆj
)
R = ( Ax + Bx ) ˆi + ( Ay + By ) ˆj = R Rx ˆi + Ry ˆj
• Rx = Ax + Bx and Ry = Ay + By R =R + R 2 x
2 y
θ= tan
−1
Ry Rx
Adding Vectors: Component Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find using components (a) A + B, (b) A − B, (c) B − A, (d) A − 2B. Report all angles counterclockwise from the positive x axis. Use a scale of:
1 unit = 0.5 m
A 3.00m cos ( 30.0° ) ˆi + 3.00m sin ( 30.0° ) ˆj A
2.60mˆi + 1.50mˆj
B = 3.00mˆj
A+ B = (2.60 + 0)mˆi + (1.50 + 3.00)mˆj A += B 2.60mˆi + 4.50mˆj = A+ B
( 2.60m )
2
2 + (4.50m) = 5.20m
−1 4.5m 60 = = θ tan 2.6m
Adding Vectors: Component Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find using components (a) A + B, (b) A − B, (c) B − A, (d) A − 2B. Report all angles counterclockwise from the positive x axis. Use a scale of:
1 unit = 0.5 m
A 3.00m cos ( 30.0° ) ˆi + 3.00m sin ( 30.0° ) ˆj A
2.60mˆi + 1.50mˆj A− B = (2.60 − 0)mˆi + (1.50 − 3.00)mˆj A −= B 2.60mˆi − 1.50mˆj
= A− B
( 2.60m )
2
+ (−= 1.50m) 2 3.00m
−1.5m tan = − 30 ,θ 2.6m −1
= 360 − 30 = 330
Adding Vectors: Component Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find using components (a) A + B, (b) A − B, (c) B − A, (d) A − 2B. Report all angles counterclockwise from the positive x axis. Use a scale of:
1 unit = 0.5 m
2.60mˆi + 1.50mˆj
A
B = 3.00mˆj
B − A = (0 − 2.60)mˆi + (3.00 − 1.50)mˆj B−A= −2.60mˆi + 1.50mˆj B−A = −1
( −2.60m )
2
+ (1.50m) 2 = 3.00m
1.5m tan 30 = − , θ = 180 − 30 = 150 -2.6m
Adding Vectors: Component Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find using components (a) A + B, (b) A − B, (c) B − A, (d) A − 2B. Report all angles counterclockwise from the positive x axis. Use a scale of:
1 unit = 0.5 m
A
2.60mˆi + 1.50mˆj
2 B = 6.00mˆj
A − 2 B= (2.60 − 0)mˆi + (1.50 − 6.00)mˆj A − 2= B 2.60mˆi − 4.50mˆj = A − 2B
4.50m) 2 5.2m ( 2.60m ) + (−= 2
−4.5m , 360 60 300 θ = − = tan −1 = − 60 2.6m
The distance between two vectors is equal to the magnitude of the difference between them! Ch3. Two points in the xy plane have Cartesian coordinates (2.00, −4.00) m and (−3.00, 3.00) m. Determine (a) the distance between these points and (b) their polar coordinates.
Problem Three displacement vectors of a croquet ball are shown, where |A| = 20.0 units, |B| = 40.0 units, and |C| = 30.0 units. Find (a) the resultant in unit-vector notation and (b) the magnitude and direction of the resultant displacement.
Rx
40.0 cos 45.0° + 30.0 = cos 45.0° 49.5
Ry
40.0sin 45.0° − 30.0sin 45.0 = ° + 20.0 27.1 = R
R=
( 49.5)
2
49.5ˆi + 27.1ˆj
+ ( 27.1) = 2
−1
27.1 θ tan = 56.4 = 49.5
28.7°
Problem A person going for a walk follows the path shown. The total trip consists of four straightline paths. At the end of the walk, what is the person's resultant displacement measured from the starting point? d1 = 100mˆi d 2 = −300mˆj d3 = −150 cos ( 30.0° ) mˆi − 150sin ( 30.0° ) mˆj = −130mˆi − 75.0mˆj d4 = −200 cos ( 60.0° ) mˆi + 200sin ( 60.0° ) mˆj = −100mˆi + 173mˆj
(
)
R =d1 + d 2 + d3 + d 4 = −130ˆi − 202ˆj m
R=
( −130 )2 + ( −202 )2 =
240 m
202 = φ tan −1 = 57.2° 130
θ = 180 + φ =
237°
Mule Problem The helicopter view shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown, and (b) the force that a third person would have to exert on the mule to make the resultant force equal to zero. The forces are measured in units of newtons (abbreviated N).
F1 120 cos ( 60.0° ) Nˆi + 120sin ( 60.0° ) Nˆj = (60.0ˆi + 104ˆj)N F2 = −80.0 cos ( 75.0° ) Nˆi + 80.0sin ( 75.0° ) Nˆj = (−20.7ˆi + 77.3ˆj)N
(
)
F = F1 + F2 = 39.3ˆi + 181ˆj N
(
)
F3 =−F = −39.3iˆ − 181ˆj N
F=
39.32 + 1812 =
181 = θ tan −1 = 39.3
185 N
77.8°SW