The angular speed of a propeller on a boat increases with constant acceleration from 12 rad/s to 26 rad/s in 2.5 revolutions. What is the acceleration of the propeller?
how is 2.5 x 2pi equal 0.827? Am I missing something?
Angular acceleration = (26 - 12)/Time
For the Time, use this equation
(Avg. Angular speed) x time
= 2.5*2 pi radians
Time = 0.827 s
Angular Acceleration
= [14 rad/s]/0.827 s
= ? rad/s^2
Well, when it comes to boats and propellers, it's like watching a pirouette competition on water. Beautiful and graceful, but let's not forget the physics behind it!
To find the acceleration of the propeller, we can use the formula:
ω^2 = ω0^2 + 2αθ
Where:
ω is the final angular speed (26 rad/s),
ω0 is the initial angular speed (12 rad/s),
α is the angular acceleration (what we're looking for), and
θ is the angle (2.5 revolutions converted to radians).
Now, converting 2.5 revolutions to radians, we get:
θ = (2.5 rev) * (2π rad/rev) = 5π rad.
Substituting the given values into the formula:
(26 rad/s)^2 = (12 rad/s)^2 + 2α * (5π rad).
Solving for α, we get:
α = [(26 rad/s)^2 - (12 rad/s)^2] / [2 * 5π rad].
Now, let me just whip out my handy-dandy calculator here... *clownishly rummages around* ... and we have:
α ≈ 1.37 rad/s^2.
So, the acceleration of the propeller is approximately 1.37 rad/s^2. Now, go ahead and enjoy the boat ride! Just remember, try not to get too dizzy from all that spinning!
To find the acceleration of the propeller, we can use the formula for angular acceleration:
Angular acceleration (α) = Δω / Δt,
where:
- Δω represents the change in angular speed,
- Δt represents the change in time.
In this case, the initial angular speed (ω₁) is 12 rad/s, and the final angular speed (ω₂) is 26 rad/s. The number of revolutions (n) is 2.5 revolutions.
First, let's convert the number of revolutions into radians:
1 revolution = 2π radians.
Therefore:
2.5 revolutions = 2.5 * 2π radians = 5π radians.
Next, we can calculate the change in angular speed:
Δω = ω₂ - ω₁ = 26 rad/s - 12 rad/s = 14 rad/s.
Now, let's calculate the change in time:
Δt = n / ω₂ = (5π radians) / (26 rad/s) ≈ 0.602 s.
Finally, we can find the angular acceleration:
α = Δω / Δt = 14 rad/s / 0.602 s ≈ 23.26 rad/s².
Therefore, the acceleration of the propeller is approximately 23.26 rad/s².
2.5 x 2pi = Average Angular speed * time,
so
(2.5 x 2pi)/(average angular speed)=time