topic 2.1 extended j – angular measure
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Topic 2.1 Extended J – Angular measure. y. x. FYI: Even though is constantly changing, the sum (cos 2 θ + sin 2 θ ) remains constant!. Topic 2.1 Extended J – Angular measure. A particle is in circular motion if it travels in a circle (or arc of a circle). - PowerPoint PPT PresentationTRANSCRIPT
Topic 2.1 ExtendedJ – Angular measure
Topic 2.1 ExtendedJ – Angular measure
A particle is in circular motion if it travels in a circle (or arc of a circle).
x
y
θr
y = r sin θx = r cos θ
The x and y coordinates of the particle at any time t can be determined using the relationships
At any time t, the angle θ is known.
The position vector r is given byr = xi + yj
r = r cos θ x + r sin θ y
How do you know that |r| is
constant?
x
y
From Pythagoras we knowr2 = x2 + y2
r2 = r2 cos2 θ + r2 sin2
θ
r2 = r2(cos2 θ + sin2
θ) r2 = r2
FYI: Even though is constantly changing, the sum (cos2 θ + sin2
θ ) remains constant!
Topic 2.1 ExtendedJ – Angular measure
θ0
t0
If we look at the particle at two different times we have two different s:
t
θWe can then define the angular displacement analogous to the linear displacement x:
= - 0 Angular Displacement
Recall that x = x - x0 told us the change in position of an object.Thus that = - 0 tells us the change in angular position of an object.
FYI: We can measure in either DEGREES or RADIANS.
FYI: It turns out that measuring angles in RADIANS simplifies our equations for circular motion greatly.
Topic 2.1 ExtendedJ – Angular measure
Do you remember how to convert from degrees to radians, or the other way around?
2 rad = 360or
rad = 180
Radian Conversion
Convert 47 into radians, and 1.29 radians into degrees.
47 rad 180 = 0.82 rad
1.29 rad 180 rad = 73.9
Topic 2.1 ExtendedJ – Angular measure
Do you remember how to find an arc length s if given a radius r and an angle ?
s = rθ, Definition of Arc Length sθ in radians
FYI: If is NOT in radians, our arc length formula becomes
s = 180rθ
θ in degrees
This is why we call RADIANS the "NATURAL" unit for angle measurement.
From the arc length formula we see that
θ = sr
=mm
...demonstrating that "radians" is really a unitless quantity.
FYI: This is yet another reason to say that the RADIAN is the natural unit for measuring angles.
FYI: In fact, we can write an angle as a pure number if we wish. Thus, we can write 2.5 radians as 2.5. Hence, we can call the unit "radians" a "ghost" unit - it appears and disappears at our own whim!
Topic 2.1 ExtendedJ – Angular measure
Suppose a 100-m tall building is located fairly far away. Suppose further that you measure it's angle of inclination to be 2. How far away is it?
θ = 2h = 100 m
r = ?
FYI: If the height h of the building is sufficiently small compared to the distance r you are from it, s = h.
s = rθ
h = rθ
r = hθ
2 rad 180 = 0.035 rad
= 100 m0.035
= 2857 m