Tangential and centripetal acceleration problem

In summary, a windmill starts from rest and has a constant angular acceleration of 0.25 rad/s^2. The tangential acceleration at the tip of the blade equals the centripetal acceleration at the same point when the angular speed is equal to the tangential acceleration divided by the radius. This can be found by setting the expressions for tangential and centripetal acceleration equal to each other and solving for the angular speed. The r's do not cancel out, but with the given information, the value of ω can be found as a function of time.
  • #1
r_swayze
66
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A windmill starts from rest and rotates with a constant angular acceleration of 0.25 rad/s2. How many seconds after starting will the magnitude of the tangential acceleration of the tip of a blade equal the magnitude of the centripetal acceleration at the same point?

I don't exactly know where to start with this problem. Wouldnt I need the radius to solve for the tangential acceleration as well as the centripetal acceleration?
 
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  • #2
It might work out without r. Give it a try! I suggest you begin with putting that condition into symbols:
tangential acceleration = centripetal acceleration
rα = v²/r (pardon the poor alpha character after the first r)
Fill in the details and see if the r's cancel out!
 
  • #3
r_swayze said:
A windmill starts from rest and rotates with a constant angular acceleration of 0.25 rad/s2. How many seconds after starting will the magnitude of the tangential acceleration of the tip of a blade equal the magnitude of the centripetal acceleration at the same point?

I don't exactly know where to start with this problem. Wouldnt I need the radius to solve for the tangential acceleration as well as the centripetal acceleration?
No. You have to work it out to see why.

What is the tangential acceleration (convert angular acceleration to tangential acceleration - use r for the radius).

Now, write out the expression for centripetal acceleration of a mass located at the tip in terms of angular speed.

At what speed does the centripetal acceleration equal the tangential acceleration?

AM
 
  • #4
I don't see how the r's can cancel out with ra = v^2/r

And how can I covert angular acceleration to tangential acceleration?
 
  • #5
It was rα = v²/r where the 2nd character is an alpha, angular acceleration.
No r's cancel at this point, but you are given that α = .25 and of course v = rω.
Since it is constant angular acceleration, you can also get a value for ω as a function of time . . . just keep working on that equation with these details and see what happens.
 

Related to Tangential and centripetal acceleration problem

1. What is tangential acceleration?

Tangential acceleration is the rate of change of an object's tangential velocity, which is the component of velocity that is parallel to the object's motion. It is typically measured in meters per second squared (m/s^2).

2. How is tangential acceleration different from centripetal acceleration?

Tangential acceleration is a measure of how an object's speed changes over time, while centripetal acceleration is a measure of how an object's direction changes over time. Tangential acceleration is always perpendicular to centripetal acceleration.

3. What is the formula for calculating tangential acceleration?

The formula for tangential acceleration is a = r * ω^2, where a is the tangential acceleration, r is the radius of the object's circular path, and ω is the angular velocity of the object. Angular velocity is the rate at which an object rotates around a fixed point.

4. How does tangential acceleration affect circular motion?

Tangential acceleration is responsible for changing the speed of an object in circular motion. As the tangential velocity increases, the object's speed increases, and as the tangential velocity decreases, the object's speed decreases. This acceleration also affects the object's centripetal acceleration, as the two are always perpendicular to each other.

5. Can tangential acceleration be negative?

Yes, tangential acceleration can be negative. This means that the object's tangential velocity is decreasing, resulting in a decrease in speed. This could happen if the object is slowing down or changing its direction of motion. However, the magnitude of tangential acceleration is always positive, as it is the rate of change of speed.

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