Anonymous
Anonymous asked in Science & MathematicsPhysics · 4 weeks ago

Is acceleration always constant with a ball rolling down an incline?

For a science project I was going to find acceleration with the kinematic equation:

v  = v0 + at

and then I was going use that value to find the friction coefficient of the ramp with:

a = (mgsinx - (friction)mgcosx) / m

However, this won't work if acceleration isn't constant right?

3 Answers

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  • Anonymous
    4 weeks ago

    acceleration is constant , provided :

    1) friction coefficient μ is constant

    2) slope angle Θ is constant

    motion with friction of a mass m

    gravitation force GF = m*g*sin Θ (down)

    friction force FF = m*g*cos Θ*μ (up)

    acceleration a = m*g*(sin Θ-cos Θ*μ)/m = g*(sin Θ-cos Θ*μ) (down)

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  • NCS
    Lv 7
    4 weeks ago

    Yes, we usually take the acceleration to be constant.

    However, the equation you provide for the acceleration is irrelevant in this case. It applies to an object SLIDING (not rolling) down the incline.

    For a rolling object, you need its moment of inertia. If the ball is solid, then

    I = (2/5)mR²

    for m, R the mass, radius.

    Then you convert its initial GPE into translational KE and rotational KE:

    mgh = ½mv² + ½Iω²

    "rolling" means that ω = v/R, so

    mgh = ½mv² + ½(2/5)mR²(v/R)² = ½mv² + (1/5)mv² = 0.7mv²

    mass m cancels, and

    v² = gh / 0.7

    from

    v² = u² + 2ad, when initial velocity u = 0, we get

    v² = 2ad

    so

    2ad = gh / 0.7

    a = gh / 1.4d

    where h is the height and d is the length of the incline.

    You could make further substitutions if you like, knowing that

    h = d*sinΘ

    If the ball is hollow, then I = (2/3)mR² and rework.

    If you find this helpful, please select Favorite Answer!

    • Dixon
      Lv 7
      4 weeks agoReport

      good point!

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  • Dixon
    Lv 7
    4 weeks ago

    Indeed. Acceleration will be constant to the extent that you can justifiably ignore air resistance. To make that true you would go for a gentle slope and a dense ball, eg a ball bearing rather than a table tennis ball. 

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