A merry-go-round accelerates from est at .870 rad/sec^2. When the merry-go-round has made 4 complete revolutions, how much time has elapsed since the merry-go round started?
What is the angular velocity of the merry-go round?
Could someone help, please
a = angular acceleration
w = a t = angular velocity
theta = (1/2) a t^2 = radians turned total from rest
in this case thetas = 2pi * 4 = 8 pi radians
so
(1/2) a t^2 = 8 pi
you know a, get t and then w
to double check , the correct answer for time is 7.60 sec? and w is 6.61 rad/s?
(.87/2)t^2 = 8 pi
t = 7.599 yes 7.6
w = .87*7.6 = 6.61 yes
To solve this problem, you need to use the kinematic equation that relates angular displacement, initial angular velocity, angular acceleration, and time:
θ = ω*t + (1/2)*α*t^2
Where:
θ is the angular displacement (in radians)
ω is the initial angular velocity (in rad/sec)
α is the angular acceleration (in rad/sec^2)
t is the time (in seconds)
First, let's solve for the angular displacement when the merry-go-round has made 4 complete revolutions. One revolution is equal to 2π radians, so 4 complete revolutions would be 4 * 2π = 8π radians.
θ = 8π radians
Next, we know that the initial angular velocity is 0 (since it starts from rest).
ω = 0 rad/sec
Finally, the given angular acceleration is 0.870 rad/sec^2.
α = 0.870 rad/sec^2
By substituting these values into the kinematic equation, we can solve for the time (t):
8π = 0 * t + (1/2) * 0.870 * t^2
8π = 0.435 * t^2
t^2 = (8π) / 0.435
t = sqrt((8π) / 0.435)
Calculating the square root, we get:
t ≈ 9.15 seconds
So, approximately 9.15 seconds have elapsed since the merry-go-round started.
To find the angular velocity at any given time, you can use the equation:
ω = ω_0 + α * t
where ω_0 is the initial angular velocity, α is the angular acceleration, and t is the time.
In this case, the initial angular velocity is 0, and the angular acceleration is 0.870 rad/sec^2. Substituting these values:
ω = 0 + 0.870 * 9.15
ω ≈ 7.96 rad/sec
Therefore, the angular velocity of the merry-go-round is approximately 7.96 rad/sec.