A helicopter propeller blade starts from rest and experiences a constant angular acceleration of 25.0 rad/s2. Find the speed of a point on the blade 4.00 m from the axis of rotation when the blade has completed 20 revolutions.
Why did the helicopter propeller blade start a revolution club? Because it wanted to experience some constant angular acceleration of fun! Now, let's calculate the speed of a point on the blade.
We first need to find the final angular velocity of the blade. We can use the equation:
ω = ω₀ + αt,
where ω is the final angular velocity, ω₀ is the initial angular velocity (which is 0 since it starts from rest), α is the angular acceleration, and t is the time.
Since the blade starts from rest, ω₀ = 0, and the equation becomes:
ω = αt.
We need to convert 20 revolutions into radians. Each revolution is equal to 2π radians, so 20 revolutions is equal to:
20 revolutions × 2π radians/revolution = 40π radians.
Now we can find the time it takes for the blade to complete 20 revolutions using the equation:
θ = ω₀t + 0.5αt²,
where θ is the angular displacement.
Since the blade starts from rest, ω₀ = 0, and the equation becomes:
θ = 0.5αt².
Substituting the given values, we have:
40π radians = 0.5 × 25.0 rad/s² × t².
Simplifying, we get:
40π = 12.5t².
Dividing both sides by 12.5, we get:
(40π) / 12.5 = t².
Solving for t, we find:
t ≈ sqrt((40π) / 12.5).
Now that we have the time it takes for the blade to complete 20 revolutions, we can find the final angular velocity. Substituting the values into ω = αt, we get:
ω = 25.0 rad/s² × sqrt((40π) / 12.5).
Finally, we can find the speed of a point on the blade using the equation:
v = ωr,
where v is the speed, ω is the angular velocity, and r is the radius.
Substituting the given radius of 4.00 m, we get:
v = (25.0 rad/s² × sqrt((40π) / 12.5)) × 4.00 m.
Calculating this expression should give you the speed of a point on the blade. Just a friendly reminder, don't try to catch that point on the blade. It's quite sharp!
To solve this problem, we'll use the following equation:
ω^2 = ω_0^2 + 2αθ
Where:
ω = final angular velocity (in rad/s)
ω_0 = initial angular velocity (0 rad/s)
α = angular acceleration (25.0 rad/s^2)
θ = angle rotated (in radians)
First, we need to convert the number of revolutions into radians:
1 revolution = 2π radians
Therefore, 20 revolutions = 20 * 2π radians = 40π radians.
Now, let's substitute the given values into the equation:
ω^2 = 0^2 + 2 * 25.0 rad/s^2 * 40π radians
ω^2 = 0 + 50.0 rad/s^2 * 40π radians
Since we know that ω = v/r (angular velocity equals linear velocity divided by radius), we can rearrange the equation as:
v = ω * r
Substituting the known values:
v = (50.0 rad/s * 4.00 m)
Calculating the value:
v = 200.0 rad·m/s
Therefore, the speed of a point on the blade 4.00 m from the axis of rotation, when the blade completes 20 revolutions, is 200.0 rad·m/s.
To find the speed of a point on the blade, we can use the equation for angular acceleration:
θ = ω₀t + 0.5αt²
where:
θ is the angle of rotation (in radians),
ω₀ is the initial angular velocity (in radians per second),
α is the angular acceleration (in radians per second squared), and
t is the time (in seconds).
First, we need to convert the angle from revolutions to radians. Since 1 revolution = 2π radians, we can multiply the given number of revolutions by 2π:
θ = 20 revolutions * 2π radians/revolution
θ = 40π radians
Next, we need to find the time it takes to complete 20 revolutions. Since the helicopter propeller blade starts from rest, the initial angular velocity (ω₀) is zero. Rearranging the equation, we get:
t = √(2θ/α)
Substituting the values, we have:
t = √((2 * 40π) / 25)
t = √(80π / 25)
t ≈ √(3.2π)
Now, we can find the final angular velocity (ω) of the point on the blade using the formula:
ω = ω₀ + αt
Since ω₀ is zero in this case:
ω = αt
ω = 25 * √(3.2π)
Finally, we can calculate the speed (v) of a point on the blade using the formula:
v = ω * r
where r is the distance from the axis of rotation. Substituting the values:
v = 25 * √(3.2π) * 4
v ≈ 314.16 m/s
Therefore, the speed of a point on the blade, located 4.00 m from the axis of rotation, when the blade has completed 20 revolutions is approximately 314.16 m/s.