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