A merry-go-round is at rest before a child pushes it so that it rotates with a constant angular acceleration for 38.0 s. When the child stops pushing, the merry-go-round is rotating at 1.20 rad/s. How many revolutions did the child make around the merry-go-round while he was pushing it?
To find the number of revolutions the child made while pushing the merry-go-round, we need to determine the angular displacement.
We can use the equation relating angular acceleration (α), initial angular velocity (ω0), time (t), and angular displacement (θ):
θ = ω0 * t + (1/2) * α * t^2
where:
θ = angular displacement
ω0 = initial angular velocity
t = time
α = angular acceleration
Given:
ω0 = 0 rad/s (since the merry-go-round was at rest before the child pushed it)
α = unknown
t = 38.0 s
θ = unknown
We also know that the final angular velocity (ω) after the child pushed the merry-go-round is 1.20 rad/s.
We can use this information to find α:
ω = ω0 + α * t
Plugging in the values:
1.20 = 0 + α * 38.0
Solving for α:
α = 1.20 / 38.0
α ≈ 0.0316 rad/s^2
Now we can substitute this value of α into the equation for θ:
θ = ω0 * t + (1/2) * α * t^2
θ = 0 * 38.0 + (1/2) * 0.0316 * (38.0)^2
θ ≈ 22.9784 rad
To find the number of revolutions, we need to convert the angular displacement to revolutions:
1 revolution = 2π radians
Number of revolutions = θ / (2π)
Number of revolutions = 22.9784 / (2π) ≈ 3.652
Therefore, the child made approximately 3.652 revolutions around the merry-go-round while pushing it.
To find the number of revolutions the child made around the merry-go-round, we need to first calculate the final angular velocity (ωf) using the initial angular velocity (ωi), the angular acceleration (α), and the time taken (t).
Given:
Initial angular velocity, ωi = 0 rad/s (since the merry-go-round was at rest)
Final angular velocity, ωf = 1.20 rad/s
Time, t = 38.0 s
To find α, we can use the formula:
ωf = ωi + αt
Substituting the given values:
1.20 rad/s = 0 rad/s + α * 38.0 s
Simplifying the equation:
1.20 rad/s = 38.0 s * α
Dividing both sides by 38.0 s:
α = 1.20 rad/s / 38.0 s ≈ 0.0316 rad/s²
Now, we can calculate the number of revolutions using the formula:
θ = ωit + (1/2)αt²
Since the initial angular velocity is zero, the equation simplifies to:
θ = (1/2)αt²
Substituting the given values:
θ = (1/2) * 0.0316 rad/s² * (38.0 s)²
Simplifying the equation:
θ = (1/2) * 0.0316 rad/s² * 1444 s²
θ ≈ 22.95 rad
To convert from radians to revolutions, we use the conversion factor:
1 revolution = 2π radians
θ_rev = θ / (2π)
θ_rev = 22.95 rad / (2π)
θ_rev ≈ 3.65 revolutions
Therefore, the child made approximately 3.65 revolutions around the merry-go-round while pushing it.
change velocity= 1.20r/s
during time=38sec
angular acceleration= 1.20/38
displacement= 1/2 angacceleration*time
displacement will be in radians. Divide that by 2PI to get revolutions.