Example Rotary Motion Problems 1. How many radians is 25 revolutions? By definition of a radian ( from C = 2 π r ) , there are 2π radians in 1 revolution 25 rev 2π rad = 50π rad 1 rev = 157 rad

Must convert 36o to radians 2. subtended angle = 36o r = 28 m L = arc length r = radius θ = # radians L θ = r L 36o Must convert 36o to radians r 36o 2π radians = 2π/10 radians 360o = 0.628 radians

2. subtended angle = 0.628 radians r = 28 m θ = L r 36o r L = r θ = ( 28 m )( 0.628 rad ) radians have no dimension, and so do not have to be canceled out; they play no role in dimensional analysis L = 17.6 m

3. r1 = 3.0 m ω1 = 4.8 rad/s r2 = 1.0 m (a) v1 = ? v = r ω v1 = r1 ω1 = ( 3.0 m )( 4.8 rad/s ) v1 = 14.4 m/s ω2 is also 4.8 rad/s (both of them make 1 revolution in the same amount of time, so their rotational speeds are equal) v2 = r2 ω2 = ( 1.0 m )( 4.8 rad/s ) = v2 = 4.8 m/s If the boy let go, he would travel at a speed of 14.4 m/s; the direction would be along a tangent to the circle

4. ω = 1500 rev/min (a) What is the period, in seconds per revolution? ω = 1500 rev/min Use dimensional analysis and conversion factors: 60 s = 1 min Goal is s/rev , given rev/min and s/min min 60 s s 0.040 = 1500 rev min rev

4. ω = 1500 rev/min (b) Convert the angular velocity to radians per second. 1500 rev 2π rad min = ω = 157 rad/s min rev 60 s (c) Find the angular displacement θ in 25 seconds. Angular velocity is constant, so use θ = ω t ( angular counterpart to d = v t ) θ = ω t = ( 157 rad/s )( 25 s ) θ = 3900 rad

5. ωi = 0 ωf = 33 rpm t = 0.50 s (a) α = ? ωf - ωi α = t Need to convert ωf to rad/s : 33 rev 2π rad 1 min = 3.46 rad/s min rev 60 s

5. ωi = 0 ωf = 3.46 rad/s t = 0.50 s (a) α = ? ωf - ωi α = t 3.46 rad/s - 0 = 0.50 s α = 6.91 rad/s2

5. ωi = 0 ωf = 3.46 rad/s t = 0.50 s α = 6.91 rad/s2 (b) θ = ? θ = ωi t + ½ α t2 = 0 + ½ ( 6.91 rad/s2 )( 0.50 s )2 θ = 0.86 rad

6. ωi = 0 t = 5.0 s α = 0.48 rad/s2 (a) ωf = ? ωf = ωi + α t = 0 + ( 0.48 rad/s2 )( 5.0 s ) ωf = 2.4 rad/s (b) θ = ? θ = ωi t + ½ α t2 = 0 + ½ ( 0.48 rad/s2 )( 5.0 s )2 θ = 6.0 rad

7. ωi = 0 α = 32.6 rad/s2 θ = 8 rev ωf = ? ωf 2 = ωi 2 + 2 α θ θ = ( 8 rev )( 2π rad/rev ) θ = 16π rad ωf 2 = 0 + 2 α θ θ = 50.3 rad ωf 2 = 2 α θ ωf 2 = 2 ( 32.6 rad/s2 )( 50.3 rad ) ωf 2 = 3277 rad2/s2 ωf = 3277 rad2/s2 ωf = 57.2 rad/s 57.2 rad rev = ωf = 9.11 rev/s s 2π rad