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Homework Help: Ball rolling down frictionless incline - Does doubling the angle double the speed?

  1. Sep 12, 2006 #1
    A ball is rolled down a frictionless incline at some angle [tex]\theta[/tex] below the horizontal. If you increase the angle of the incline by a factor of two (make the ramp steeper downward by twice as much), does the ball roll down at double the speed?

    Here is what I said:

    No. because of the following relation

    [tex] \[
    \begin{array}{l}
    Speed = \sqrt {\left( {v_i \cos \left( \theta \right)t} \right)^2 + \left( {h + v_i \sin \left( \theta \right)t + \frac{1}{2}gt^2 } \right)^2 } \\
    2Speed \ne \sqrt {\left( {v_i \cos \left( {2\theta } \right)t} \right)^2 + \left( {h + v_i \sin \left( {2\theta } \right)t + \frac{1}{2}gt^2 } \right)^2 } \\
    \end{array}
    \]
    [/tex]

    Where [tex]v_i[/tex] is the initial velocity of the ball, [tex]\theta[/tex] is the angle of the ramp below the horizontal, [tex]h[/tex] is the initial height of the ball, [tex]g[/tex] is the acceleration due to gravity, and [tex]t[/tex] time.


    because

    [tex]
    \[
    \begin{array}{l}
    \cos (\theta ) \ne \cos (2\theta ) \\
    \sin (\theta ) \ne \sin (2\theta ) \\
    0 < \theta < \frac{\pi }{2} \\
    \end{array}
    \]
    [/tex]


    Is this a correct way to show that increasing the downward angle by a factor of two does not double the speed of the object rolling down the incline?
     
    Last edited: Sep 12, 2006
  2. jcsd
  3. Sep 12, 2006 #2

    Doc Al

    User Avatar

    Staff: Mentor

    Why not just write an expression for the acceleration along the incline as a function of angle? (Are you concerned with something rolling, or did you just mean sliding without friciton?)

    I assume you mean speed as a function of time as it goes down the incline. The speed at the bottom will only depend on the height.
     
  4. Sep 12, 2006 #3
    It is speed as a function of time, and I guess the object could be rolling or sliding with no resistance.

    Even though my answer isn't the simpliest, is it still a correct way to show that increasing the downward angle by a factor of two does not double the speed?

    What is the simplest way to show that the increasing the downward slope of the ramp by a factor of two does not double the speed?
     
  5. Sep 12, 2006 #4

    Doc Al

    User Avatar

    Staff: Mentor

    I don't understand your equation, since you have speed on one side but distance on the other.

    Imagine a block sliding down a frictionless slope making an angle [itex]\theta[/itex] with the horizontal. What's its acceleration down the incline?
     
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