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lapo4ka [179]
3 years ago
12

Please answer the question below (ABOUT VECTORS AND MAGNITUDE)

Mathematics
1 answer:
coldgirl [10]3 years ago
4 0

(a) <em>v</em> appears to have a fixed direction along the positive <em>x</em>-axis. If ||<em>u</em>|| = 150 N, ||<em>v</em>|| = 220 N, then when <em>θ</em> = 30°, you have

<em>u</em> = (150 N) (cos(30°) <em>i</em> + sin(30°) <em>j</em> ) ≈ (129.904 <em>i</em> + 75 <em>j</em> ) N

<em>v</em> = (220 N) (cos(0°) <em>i</em> + sin(0°) <em>j</em> ) = (220 <em>i</em> ) N

(<em>i</em> and <em>j</em> are the unit vectors in the positive <em>x</em> and <em>y</em> directions)

and their sum is

<em>u</em> + <em>v</em> ≈ (349.904 <em>i</em> + 75 <em>j</em> ) N

with magnitude

||<em>u</em> + <em>v</em>|| ≈ √((349.904)² + (75)²) N ≈ 357.851 N ≈ 357.9 N

and at angle <em>φ</em> made with the positive <em>x</em>-axis such that

tan(<em>φ</em>) ≈ (75 N) / (349.904 N)   →   <em>φ</em> ≈ 12.098° ≈ 12.1°

(b) Letting <em>θ</em> vary from 0° to 180° would make <em>v</em> a function of <em>θ</em> :

<em>u</em> = (150 N) (cos(<em>θ</em>) <em>i</em> + sin(<em>θ</em>) <em>j</em> ) = (150 cos(<em>θ</em>) <em>i</em> + 150 sin(<em>θ</em>) <em>j</em> ) N

Then

<em>u</em> + <em>v</em> = ((220 + 150 cos(<em>θ</em>)) <em>i</em> + (150 sin(<em>θ</em>)) <em>j</em> ) N

→   <em>M</em> = ||<em>u</em> + <em>v</em>|| = √((220 + 150 cos(<em>θ</em>))² + (150 sin(<em>θ</em>))²) N

<em>M</em> = √(48,400 + 66,000 cos(<em>θ</em>) + 22,500 cos²(<em>θ</em>) + 22,500 sin²(<em>θ</em>)) N

<em>M</em> = 10 √(709 + 660 cos(<em>θ</em>)) N

(c) As a function of <em>θ</em>, <em>u</em> + <em>v</em> makes an angle <em>α</em> with the positive <em>x</em>-axis such that

tan(<em>α</em>) = (150 sin(<em>θ</em>) / (220 + 150 cos(<em>θ</em>))

→   <em>α</em> = tan⁻¹((15 sin(<em>θ</em>) / (22 + 15 cos(<em>θ</em>)))

(d) Filling in the table is just a matter of evaluating <em>M</em> and <em>α</em> for each of the given angles <em>θ</em>. For example, when <em>θ</em> = 0°,

<em>M</em> = 10 √(709 + 660 cos(0°)) N = 370 N

<em>α</em> = tan⁻¹((15 sin(0°) / (22 + 15 cos(0°))) = 0°

When <em>θ</em> = 30°, you get the same result as in part (a).

When <em>θ</em> = 60°,

<em>M</em> = 10 √(709 + 660 cos(60°)) N ≈ 323.3 N

<em>α</em> = tan⁻¹((15 sin(60°) / (22 + 15 cos(60°))) = 23.8°

and so on.

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