Answer:
a) α = 0.375 rad/s²
b) at = 0.1875 m/s²
c) ac =79 m/s²
d) θ = 52 rad
Explanation:
The uniformly accelerated circular movemeis a circular path movement in which the angular acceleration is constant.
Tangential acceleration is calculated as follows:
at = α*R Formula (1)
Centripetal acceleration is calculated as follows:
ac =ω² *R Formula (2)
We apply the equations of circular motion uniformly accelerated :
ωf= ω₀ + α*t Formula (3)
θ= ω₀*t + (1/2)*α*t² Formula (4)
Where:
θ : angle that the body has rotated in a given time interval (rad)
α : angular acceleration (rad/s²)
t : time interval (s)
ω₀ : initial angular speed ( rad/s)
ωf : final angular speed ( rad/s)
R : radius of the circular path (m)
at: tangential acceleration, (m/s²)
ac: centripetal acceleration, (m/s²)
Data:
R= 0.5 m : radius of the disk
t₀=0 , ω₀ = 5 rev/s
1 revolution = 2π rad
ω₀ = 5*(2π)rad/s =10π rad/s = 31.42 rad/s
ωf = 2*(2π)rad/s =4π rad/s = 12.57 rad/s
t = 8 s
(a) angular acceleration of the box
We replace data in the formula (3)
ωf= ω₀ + α*t
2 = 5 + α*(8)
2 -5 = α*(8)
-3 = (8)α
α=3 /8
α = 0.375 rad/s²
(b) Tangential acceleration of the box
We replace data in the formula (1)z
at =(α)*R
at = (0.375)*(0.5)
at = 0.1875 m/s²
c) Centripetal acceleration of the box at t = 8 s
We replace data in the formula (2)
ac =ω² *R
ac =(12.57)² *(0.5)
ac = 79 m/s²
d) Radians that the box has rotated over after t = 8 s
We replace data in the formula (4)
θ = ω₀*t + (1/2)*α*t²
θ = (5)*(8)+ (1/2)*( 0.375)*(8)²
θ = 52 rad
