Find Angular Speed: Airplane Propeller Blades & Constant Acceleration

In summary, the propeller blades of an airplane, measuring 4.0 m in length, start turning from rest at a constant angular acceleration as the plane prepares for takeoff. Between the fifth and eighth second of rotation, the blades complete two full revolutions. To find the angular speed at the end of 8.2 s, we can use the formula ω = ωο + αt and the information that the total angle turned is 4PI radians. Solving for α and multiplying it by 8.2, we get an angular speed of approximately 5.28, as confirmed by the textbook.
  • #1
nesan
75
0

Homework Statement


The propeller blades of an airplane are 4.0 m long. The plane is getting ready for takeoff, and the propeller starts turning from rest at a constant angular acceleration. The propeller blades go through two revolutions between the fifth and the eighth second of the rotation. Find the angular speed at the end of 8.2 s.

The Attempt at a Solution


v = r ω

It seems very easy but I'm stuck on how to find ω

I know there's a constant acceleration

so

ω = ωο + αt

Can someone point me in the right direction with how oto use

"The propeller blades go through two revolutions between the fifth and the eighth second of the rotation."

to get the acceleration.

Than you.
 
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  • #2
nesan said:
ω = ωο + αt

What is the angular velocity when t = 0, i.e., ωο? Given this, what is the total angle turned as a function of time?
 
  • #3
Orodruin said:
What is the angular velocity when t = 0, i.e., ωο? Given this, what is the total angle turned as a function of time?
Since it says it starts at rest, when t = 0, ωο would be 0?

We use the other formula

θ = ωot + 1/2 αt^2

So θ(t) = 1/2αt^2

How would I figure out α?
 
  • #4
There is some information about ##\theta(t)## given in the problem formulation. Can you decipher it?
 
  • #5
Orodruin said:
There is some information about ##\theta(t)## given in the problem formulation. Can you decipher it?
Whoohoo, I got it.

"The propeller blades go through two revolutions between the fifth and the eighth second of the rotation."

So

θ(8) - θ(5) = 4 PI

- > α (0.5 * 82 - 0.5 * 52) = 4 PI

Solve for α and times it by 8.2 to get angular speed.

I got approximately 5.28 which my textbook says is correct. :)

Thank you so much Orodruin.
 

Related to Find Angular Speed: Airplane Propeller Blades & Constant Acceleration

1. What is angular speed?

Angular speed is a measure of how fast an object is rotating around a fixed point. It is often measured in radians per second (rad/s) or revolutions per minute (RPM).

2. How is angular speed related to linear speed?

Angular speed and linear speed are related by the formula v = rω, where v is linear speed, r is the distance from the object to the center of rotation, and ω is angular speed. This means that the faster an object's angular speed, the faster it will be moving in a circular motion.

3. How does a constant acceleration affect angular speed?

If an object experiences a constant acceleration, its angular speed will also change at a constant rate. This means that the object will either speed up or slow down depending on the direction of the acceleration.

4. How can I calculate the angular speed of an airplane propeller blade?

The angular speed of an airplane propeller blade can be calculated by dividing the number of revolutions per minute (RPM) by 60 to get the number of revolutions per second, and then multiplying by 2π to convert to radians per second (rad/s).

5. What factors can affect the angular speed of an airplane propeller blade?

The angular speed of an airplane propeller blade can be affected by the RPM of the engine, the length and shape of the propeller blade, and the air resistance and drag on the blade. Other factors such as altitude and air density can also play a role in the propeller's angular speed.

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