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Energy problem

  1. Mar 20, 2015 #1
    1. The problem statement, all variables and given/known data
    nVKXhyn.png

    2. Relevant equations
    1/2 mv^2 = mgh

    3. The attempt at a solution
    first, a side question, if p = mv, can't we just place p into 1/2 mv^2 so it becomes 1/2 p^2?

    anyway, on to the question, at the point where the cube falls off, the Fnormal should just about equal 0, correct?
    is this the way to approach the problem? Or we can find the velocity where the cube loses contact with the sphere and plug that v into 1/2 mv^2 = mgh, solving for h.
     
    Last edited: Mar 20, 2015
  2. jcsd
  3. Mar 20, 2015 #2
    3. No you can't because there is only v square while m isn't.
    Personally I would prefer to consider the balance of two force, gravity and centripetal, looking at which speed the mass can still move of a rotational motion.
     
  4. Mar 20, 2015 #3
    so at the point the cube loses contact with the half-sphere, acceleration becomes 9.8 m/s^2.

    a = v^2/r

    9.8 x 10.3m = v^2
    v= 10.04 m/s

    1/2 m v^2 = mgh
    mass cancels out

    1/2 (10.04)^2 = g h

    h = 5.15 meters.

    is that correct?
     
  5. Mar 20, 2015 #4
    When the ball will freely fall then it's total acceleration will be ##g##. You just need here radial acceleration.

    ccc.png

    If the block looses contact with the sphere then what can you say about the magnitude of Normal reaction force acting on the block.
    If the block has velocity at ##v## at the instant shown then what should be the centripetal force acting on it? This centripetal force it provided by which force? Equate them and get the answer.
     
  6. Mar 20, 2015 #5
    Centripetal force is provided by Fnormal? which is 0 when it loses contact with the half-sphere.

    correct?
     
  7. Mar 20, 2015 #6
    Yes Normal force will be zero when it looses contact. But centripetal force is provided by the component of mg. Can you find it?
     
  8. Mar 20, 2015 #7
    Centri. Force = m a

    a = v^2/r = mg ?
     
  9. Mar 20, 2015 #8
    I said a ##component## of mg. Can you find it in terms of mg and ##\theta##?
     
  10. Mar 20, 2015 #9
    it should be the hypotenuse in the picture, which is mg / cos theta

    how do we solve for theta?
     
  11. Mar 20, 2015 #10
    It will be ##mgcos\theta##

    First equate this to centripetal force. You can find relation between ##v## and ##\theta## by conservation of energy.
     
  12. Mar 20, 2015 #11
    mgcosθ = m v^2 / r


    gcosθ = v^2/r

    correct?
     
  13. Mar 20, 2015 #12
    Yes. Now use conservation of energy to find relation between v and theta.
     
  14. Mar 20, 2015 #13
    conservation of energy = 1/2 mvi^2 + mghi = 1/2 mvf^2 + mghf ?
     
  15. Mar 20, 2015 #14
    What is the initial velocity of the block?
    Can you represent hi and hf in terms of R and theta?
     
  16. Mar 20, 2015 #15
    initial velocity should be 0 since it was at rest.

    hi is 10.3m

    hf is what we are looking for

    and i have no idea how to find theta.
     
  17. Mar 20, 2015 #16
  18. Mar 20, 2015 #17
    hf is ##Rcos\theta##. Now proceed.
     
  19. Mar 20, 2015 #18
    but we have 2 unknowns, hf and theta.
     
  20. Mar 20, 2015 #19
    ##h_{f}=Rcos\theta##. Put this in energy equation. You would have two unknowns v and ##\theta##. Remember the equation which you made by using centripetal force. Use these two equation and find ##\theta##.:smile:
     
  21. Mar 20, 2015 #20
    is v = 14 m/s?
     
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