Vector Components. Calculator Trig Functions Make sure calculator is in DEG NOT RAD or GRAD (turn...
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Transcript of Vector Components. Calculator Trig Functions Make sure calculator is in DEG NOT RAD or GRAD (turn...
Practice: find the followingcos 30ºsin 30ºcos 60ºsin 60ºcos 45ºsin 45ºcos 23ºsin 23º
0.8660.5000.500 0.8660.707 0.707 0.921 0.391
Vector ComponentsAny vector is the vector sum of an x-
component and a y-componenta = ax + ay
a
ax
ayy
x
θ
ExampleResolve a vector of 5 cm at 60º into it’s x and
y components:ay = a sinθ ax = a cosθ
= 5 sin 60º = 5 cos 60ºay = 4.33 ax = 2.5
note: sin 60º = 0.866 cos60º = 0.5
ExampleResolve 10 cm at 45º into x and y
ay = a sinθ ax = a cosθ
= 10 sin 45º = 10 cos 45ºay = 7.07 cm ax = 7.07 cm
ExampleResolve a vector of 4 m/s at 143º
ay = a sinθ ax = a cosθ= 4 sin 143º = 4 cos 143º
ay = 2.41 m/s ax = -3.19 m/s
note: sin 143º = 0.602 cos 143º = -0.799 why negative?
ExampleResolve a vector of 92 m/s2 at 230º
ay = a sinθ ax = a cosθ= 92 sin 230º = 92 cos 230º
ay = - 70.5 m/s ax = -59.1 m/s
note: sin 230º = -0.766 cos 230º = -0.642
ExampleYou walk northeast 4000 m. (That’s 45º
north of east). How far east and how far north have you walked?
N
W
S
E
4000 m
45º
East: = 4000 cos 45= 4000 x 0.707= 2828 m
North: = 4000 sin 45= 4000 x 0.707= 2828 m
2828m N
2828m E
4000 m
45º
Adding Vectors w/ Components1. Find x and y components of all vectors2. Add all x-components3. Add all y-components4. You now have rx and ry
5. Use Pythag. to find magnitude6. Use θ = tan-1(ry/rx) to find the angle
Example: Add a and ba = 3.0 N @ 20º b = 2.0 N @ 45º
ax = 3 cos 20º bx = 2 cos 45º
= 2.82 N (x) = 1.41 N (x)ay = 3 sin 20º by = 2 sin 45º
= 1.03 N (y) = 1.41 N (y)
Add x and y componentsax + bx =2.82 N + 1.41 N
= 4.23 N (x) = rx
ay + by = 1.03 N + 1.41 N
= 2.44 N (y) = ry
Use Pythag. to find resultant rr2 = (rx)2 + (ry)2
r2 = (4.23)2 + (2.44)2
r2 = 23.85r = 4.88 N
rx = 4.23
ry = 2.44
r
θ
Use θ = tan-1(ry/rx) to find angleθ = tan-1(ry/rx)
= tan-1(2.44/4.23)= tan-1 (0.577)
θ = 30º
rx = 4.23
ry = 2.44r
θ
a = 4.0 m/s @ 35º b = 5.6 m/s @ 140º
ax = 3.28 m/s bx = -4.29 m/s
ay = 2.29 m/s by = 3.60 m/s
ax + bx = -1.01 m/s (x) = rx
ay + by = 5.89 m/s (y)= ry
ab +y
+x-xExample 2
r2 = (-1.01)2 + (5.89)2
r2 = 35.71r = 5.98 Nθ = tan-1(ry/rx)
= tan-1(5.89/-1.01)= tan-1 (-5.83)
θ = -80.3º ???θ = 180 + -80.3º = 99.7
rx = -1.01
ry = 5.89
r = 5.98
θ