Newton's Laws Applications

Question

Helicopter blades withstand tremendous stresses. In addition to supporting the weight of a helicopter, they are spun at rapid rates and experience large centripetal accelerations, especially at the tip. (For each answer, enter a number.)

    

Calculate the magnitude (in m/s²) of the centripetal acceleration at the tip of a 4.40 m long helicopter blade that rotates at 290 rev/min.

__ m/s²

    

Compare the linear speed of the tip with the speed of sound (taken to be 340 m/s).

v_tip/v_sound = __

Answer 1

1. Centripetal acceleration can be calculated using the formula a_c =v²/R
2. Here they tell us to use R = 4.40 m, but we need to find v_tip in order to plug in.
3. We have ω = 230 rev/min
4. We'll need to convert this into rad/s (SI units) first, then use v = ωr to find the linear speed at the tip
5. 230 rev/min * (2*pi radians/ 1rev) = 460*pi radians/min
6. 460*pi radians/min * (1 min/60s) = 7.67*pi rad/s
7. Now we can use v = ωr = (7.67*pi rad/s)*4.40 m
8. I'll leave that calculation to you.
9. The last step for finding the centripetal acceleration is to return to a_c =v²/R

For part b, just use the value for v that you found towards the end there and plug in. You should get an answer under 1, as helicopter blades are designed to not break the sound barrier.