Answer:
a) 251.36 s b) -4.77 × 10⁻³ rad/s² c) 73.64 s
Step-by-step explanation:
Using equation of angular motion
θ = (ω₀+ ω₁)t/2
2θ = (ω₁ + ω₀)t where ω₀ is the initial angular speed = 1.2 rad/s, ω₁ is the final angular speed = 0 and t is the time
2 × 24 × 2π = ( 0 + 1.2) t
t = 301.632 / 1.2 = 251.36 s
b) the angular acceleration can be calculated by:
ω₁ = ω₀ + αt
ω₁ -ω₀ = αt
0 - 1.2 = 251.36 α
α = -1.2 / 251.36 = -4.77 × 10⁻³ rad/s²
c) the amount of time required to complete the first 12 of the 24 revolutions
θ = ω₀t + 0.5αt²
12 × 2π = 1.2t - 0.00239t²
75.408 = 1.2t - 0.00239t²
- 0.00239t²+ 1.2t - 75.408 = 0
multiply by -1 both sides
0.00239t²- 1.2t + 75.408 = 0
using quadratic formula to solve for t
-b ±√(b² - 4ac) / 2a
1.2 ±√( 1.44 - 0.721) / (2×0.00239)
1.2 ± √0.719 / 0.00478
1.2 - 0.848 / 0.00478 or 1.2 + 0.848 / 0.00478
0.352 / 0.00478 or 2.048 / 0.00478
t = 73.22 s or 428.5
t = 73.64 s