Topic 2.1 Extended J – Angular measure

1. Topic 2.1 ExtendedJ – Angular measure

2. y x FYI: Even though  is constantly changing, the sum (cos2θ + sin2θ ) remains constant! Topic 2.1 ExtendedJ – Angular measure A particle is in circular motion if it travels in a circle (or arc of a circle). At any time t, the angle θ is known. The x and y coordinates of the particle at any time t can be determined using the relationships r y x = rcosθ θ y = rsinθ x The position vector r is given by r = xi + yj r = rcosθx + rsinθy From Pythagoras we know How do you know that |r| is constant? r2 = x2 + y2 r2 = r2 cos2θ + r2 sin2θ r2 = r2(cos2θ + sin2θ) r2 = r2

3. Angular Displacement Topic 2.1 ExtendedJ – Angular measure If we look at the particle at two different times we have two different s: t  t0 θ We can then define the angular displacement analogous to the linear displacement x: θ0 =  - 0 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.

4. Radian Conversion Topic 2.1 ExtendedJ – Angular measure Do you remember how to convert from degrees to radians, or the other way around? 2 rad = 360 or  rad = 180 Convert 47 into radians, and 1.29 radians into degrees. 47  rad 180 = 0.82 rad 180  rad 1.29 rad = 73.9

5. Definition of Arc Length s 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 Do you remember how to find an arc lengths if given a radius r and an angle ? s = rθ, θ in radians FYI: If  is NOT in radians, our arc length formula becomes 180rθ  s = θ in degrees This is why we call RADIANS the "NATURAL" unit for angle measurement. From the arc length formula we see that m m s r θ= = ...demonstrating that "radians" is really a unitless quantity.

6. FYI: If the height h of the building is sufficiently small compared to the distance r you are from it, s = h. 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? h = 100 m θ= 2 r= ? s = rθ 2  rad 180 = 0.035 rad h = rθ h θ 100 m 0.035 = 2857 m r= =