Vector a Overscript right-arrow EndScripts has a magnitude of 5.20 m and is directed east. Vector b Overscript right-arrow EndScripts has a magnitude of 3.70 m and is directed 36.0° west of north. What are (a) the magnitude and (b) the direction (counterclockwise from east) of a Overscript right-arrow EndScripts plus b Overscript right-arrow EndScripts? What are (c) the magnitude and (d) the direction (counterclockwise from east) of b Overscript right-arrow EndScripts minus a Overscript right-arrow EndScripts?
33. An automobile tire is rated to last for 50,000 miles. Estimate the number o revolutions the tire will make in its lifetime.
36o W. of N. = 90+36 = 126o CCW from E.
(a+b) = 5.2 + 3.70[126o].
(a+b) = 5.2 + 3.70*Cos126+3.70*sin126,
(a+b) = 5.2 - 2.17+2.99i,
(a+b) = 3.03 + 2.99i = 4.26m[44.62o]
A. Magnitude = 4.26 m.
B. Direction = 44.62o.
(b-a) = 3.70[126o] - 5.2.
(b-a) = -2.17+2.99i - 5.2,
(b-a) = -7.37 + 2.99i = 7.95m[-22.08] = 7.95m[158o].
C. Magnitude = 7.95 m.
D. Direction = 158o.
C = pi * Diameter = Circumference.
We need to know the radius or diameter of the tire.
Rev. = d/C.
d = 50,000 Miles.
d and C must be in the same unit(meters or miles).
To solve this problem, we can use vector addition and subtraction. Let's start by finding the magnitude and direction of the sum of vectors a and b.
(a) Magnitude of a + b:
To find the magnitude of the sum, we can use the Pythagorean theorem. The sum of two vectors can be found by:
|a + b| = sqrt(|a|^2 + |b|^2 + 2|a||b|cosθ)
Using the given magnitudes:
|a| = 5.20 m
|b| = 3.70 m
And the angle between the vectors:
θ = 36.0°
Plugging in the values:
|a + b| = sqrt((5.20)^2 + (3.70)^2 + 2(5.20)(3.70)cos(36.0°))
Calculating this expression will give you the magnitude of vector a + b.
(b) Direction of a + b:
To find the direction of vector a + b, we need to find the angle it makes with the east direction (counterclockwise from east).
We can use the law of cosines to find this angle:
cosϕ = (|b|^2 + |a + b|^2 - |a|^2) / (2|b||a + b|)
Using the given magnitudes and the calculated magnitude of a + b, you can calculate the value of cosϕ. Then use the inverse cosine function (arccos) to find the angle ϕ. This angle will give you the direction of vector a + b.
Moving on to finding the magnitude and direction of the difference b - a:
(c) Magnitude of b - a:
The magnitude of the difference between vectors b and a can be found using the same formula as vector addition:
|b - a| = sqrt(|b|^2 + |a|^2 - 2|a||b|cosθ)
Using the given magnitudes and angle:
|a| = 5.20 m
|b| = 3.70 m
θ = 36.0°
Plugging in the values, calculate the expression to find the magnitude of vector b - a.
(d) Direction of b - a:
Similar to finding the direction of vector a + b, we can find the angle that vector b - a makes counterclockwise from east.
Use the law of cosines:
cosϕ = (|b|^2 + |b - a|^2 - |a|^2) / (2|b||b - a|)
Calculate the value of cosϕ using the magnitudes and the calculated magnitude of b - a. Then use the inverse cosine function (arccos) to find the angle ϕ, which gives you the direction of vector b - a.
By following these steps, you can calculate the magnitude and direction of both a + b and b - a.