Solve Angular Acceleration for Blade Radius | Homework Help

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In summary, the equation aT=(r)(\alpha) is used to find the radius at which the tangential acceleration is equal to the acceleration due to gravity. When rearranged, the resulting value may be in m/rad, but radians are actually dimensionless and the answer is still correct.
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Homework Statement



A fan blade is rotating with a constant angular acceleration of +12.0 rad/s2. At what point on the blade, as measured from the axis of rotation, does the magnitude of the tangential acceleration equal that of the acceleration due to gravity?

Homework Equations



aT=(r)([itex]\alpha[/itex])

The Attempt at a Solution



What I did is rearrange the equation so that r=aT/[itex]\alpha[/itex]
This gave me r=(9.80m/s^2)/(12.0rad/s^2)
So, r=0.0817 m/rad

The answer is supposed to be 0.817 m, but why am I getting m/rad? In fact, m/rad doesn't even make sense. Is there another way to rearrange the equation?

Any help is much appreciated. Thank you.
 
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Radians, although a unit of measure, are dimensionless. You are getting the correct answer using a correct method.

A similar situation is in the arc length of a circle. s = rθ has the same issue where s is the arc length (meters), r is the radius (meters) and θ is the subtended angle (radians). In a circle, θ is defined by this formula as being the arc length divided by the radius. Being two distance measurements, the units cancel to be dimensionless.

We use the term radians to differentiate it from other angle measurements like degrees which are measured as being the ratio s/r = 1/360.

While looking up information that would help explain this, I found a similar question someone asked.

I suggest reading it to further your understanding of angle measurements: http://mathforum.org/library/drmath/view/64034.html
 
Last edited:

1. What is angular acceleration?

Angular acceleration is a measure of how fast the angular velocity of an object changes over time. It is given in units of radians per second squared (rad/s^2) and is represented by the symbol α.

2. How do you calculate angular acceleration?

Angular acceleration can be calculated by dividing the change in angular velocity by the time interval over which it occurred. In other words, α = (Δω) / (Δt).

3. What is the formula for calculating blade radius using angular acceleration?

The formula for calculating blade radius using angular acceleration is r = (ω^2) / α, where r is the radius of the blade, ω is the initial angular velocity, and α is the angular acceleration.

4. How is angular acceleration related to tangential acceleration?

Angular acceleration and tangential acceleration are related through the formula at = rα, where at is the tangential acceleration and r is the distance from the center of rotation. This means that as the angular acceleration increases, the tangential acceleration also increases.

5. What factors can affect the angular acceleration of a rotating object?

The angular acceleration of a rotating object can be affected by various factors such as the torque applied to the object, the mass and distribution of the object, and the moment of inertia (i.e. how the mass is distributed around the axis of rotation). Friction and air resistance can also play a role in the angular acceleration of an object.

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