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Problem using centripital accel... Can you check my work?

  1. Mar 18, 2017 #1
    1. The problem statement, all variables and given/known data

    A student ties a 500 g rock to a 1.0-m-long string and swings it around her head in a horizontal circle.
    At what angular velocity in rpm does the string tilt down at a 16 degree angle?

    2. Relevant equations

    F = (m*v^2)/r
    v = r*w

    3. The attempt at a solution

    First I solved for the tension in the string...

    ∑Fy = 0

    T*sin(16°) - mg = 0
    T*sin(16°) = mg
    T = 17.776 N

    Next I solved for the linear velocity

    ΣFx = m*a

    Tx = 17.776cos(16°)
    r = sin(74°)
    17.776cos(16°) = ((.5kg)*v^2)/(sin(74°)
    Solve for linear velocity --> 5.73 m/s

    v = r*w
    Solve for omega and I got 5.96 rad/s

    Convert to rpm and I got 56.9 rpm.
     
  2. jcsd
  3. Mar 18, 2017 #2

    mfb

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    2016 Award

    Staff: Mentor

    I get the same values.
     
  4. Mar 18, 2017 #3
    Awesome, thank you!
     
  5. Mar 18, 2017 #4

    gneill

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    Staff: Mentor

    You've found a valid solution, well done. Since you've arrived at the answer I'm free to show you an alternative method.

    If you make a drawing of the scenario and consider the accelerations experienced by the rock you'll see that the ratio of the gravitational acceleration to the centripetal acceleration will be equal to the tan of the angle. In this diagram the negative of the centripetal acceleration (red vector) is shown for clarity:

    upload_2017-3-18_20-7-25.png

    So that ##tan(θ) = \frac{g}{ω^2 r}##, where: ##r = L cos(θ)##. With a bit of rearranging and simplifying this becomes:

    ##ω = \sqrt{\frac{g}{L sin(θ)}}##

    which should yield the same value for ω that you found.
     
  6. Mar 18, 2017 #5
    Thank you! I'll remember that :)
     
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