Ceiling Fan Blade Acceleration and Angle Calculation

[eiriyuki]

1. The blades of a ceiling fan have a radius of 0.318 m and are rotating about a fixed axis with an angular velocity of +1.48 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of +2.22 rad/s2. After 0.394 s have elapsed since the switch was reset, what is (a) the total acceleration (in m/s2) of a point on the tip of a blade and (b) the angle between the total acceleration and the centripetal acceleration (See Figure)?

http://edugen.wiley.com/edugen/courses/crs2216/art/qb/qu/c08/w1590int.gif

Homework Equations

I tried using equation $$\omega$$$$_{2}$$ = $$\omega$$$$^{2}_{o}$$ + 2$$\alpha$$$$\theta$$

The Attempt at a Solution

This is the first time I have used the forums and I'm having a hard time putting in equations and things, so I apologize if the format is silly. I solved for theta in the above equation and my answers were 3.6509 (part a) rev and 14.153 (part b). I do not know if this is right, and if it is, I am not sure how to convert it into degrees so I can figure the position on the wheel.

 

[rock.freak667]

This is the first time I have used the forums and I'm having a hard time putting in equations and things, so I apologize if the format is silly. I solved for theta in the above equation and my answers were 3.6509 (part a) rev and 14.153 (part b).

Why did you solve for θ? You don't have ω₂. You need to find ω₂.

Use the other equations of rotational motion.

Also, how does the tangential acceleration a_T relate to the angular acceleration α?

How does the centripetal acceleration relate to ω₂?

 

[eiriyuki]

Wow, that is silly. I put my work for another problem I was having trouble with.

For this problem I used, total a=$$\sqrt{(rw^{2})^{2} + (r\alpha)^{2}}$$

I worked out the problem and got 0.99m/s$$^{2}$$, which was incorrect.

 

[rock.freak667]

What value did you use ω?

 

[eiriyuki]

What value did you use ω?

1.48 rad/s

 

[rock.freak667]

1.48 rad/s

That's where you went wrong. You need to use the final angular velocity. It started out at 1.48 rad/s and then allowed to accelerate at 2.22 rad/s for 0.394s. How would you find the final angular velocity given the information?

 

[eiriyuki]

That's where you went wrong. You need to use the final angular velocity. It started out at 1.48 rad/s and then allowed to accelerate at 2.22 rad/s for 0.394s. How would you find the final angular velocity given the information?

Would I use average angular acceleration?

w - w₀ / t - t₀

Which gives me 2.35 rad/s.

 

[rock.freak667]

Now try it back in your equation.

 

[eiriyuki]

I got the correct answer for both parts now! Thank you so much for you help! :)

 

[technomaze]

Use the other equations of rotational motion.

Also, how does the tangential acceleration aT relate to the angular acceleration α?

How does the
ceiling fan relate to ω2?