Rank the following objects by their accelerations down an incline (assume each object rolls without slipping) from least to greatest:

Question

Rank the following objects by their accelerations down an incline (assume each object rolls without slipping) from least to greatest:

a. Hollow Cylinder
b. Solid Cylinder
c. Hollow Sphere
d. Solid Sphere

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Thành Đạt 2 weeks 2021-08-29T19:14:47+00:00 1 Answers 0 views 0

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    2021-08-29T19:16:22+00:00

    Answer:

    acceleration are

         hollow cylinder < hollow sphere < solid cylinder < solid sphere

    Explanation:

    To answer this question, let’s analyze the problem. Let’s use conservation of energy

    Starting point. Highest point

              Em₀ = U = m g h

    Final point. To get off the ramp

              Em_f = K = ½ mv² + ½ I w²

    notice that we include the kinetic energy of translation and rotation

             

    energy is conserved

            Em₀ = Em_f

            mgh = ½ m v² +1/2 I w²

    angular and linear velocity are related

             v = w r

             w = v / r

    we substitute

              mg h = ½ v² (m + I / r²)

              v² = 2 gh   \frac{m}{m+ \frac{I}{r^2} }

              v² = 2gh    \frac{1}{1 + \frac{I}{m r^2} }

    this is the velocity at the bottom of the plane ,, indicate that it stops from rest, so we can use the kinematics relationship to find the acceleration in the axis ax (parallel to the plane)

             v² = v₀² + 2 a L

    where L is the length of the plane

             v² = 2 a L

             a = v² / 2L

    we substitute

             a = g \ \frac{h}{L} \  \frac{1}{1+ \frac{I}{m r^2 } }

    let’s use trigonometry

             sin θ = h / L

             

    we substitute

             a = g sin θ   \ \frac{h}{L} \  \frac{1}{1+ \frac{I}{m r^2 } }

    the moment of inertia of each object is tabulated, let’s find the acceleration of each object

    a) Hollow cylinder

          I = m r²

    we look for the acerleracion

          a₁ = g sin θ    \frac{1}{1 + \frac{mr^2 }{m r^2 } }1/1 + mr² / mr² =

          a₁ = g sin θ    ½

    b) solid cylinder

           I = ½ m r²

           a₂ = g sin θ  \frac{1}{1 + \frac{1}{2}  \frac{mr^2}{mr^2} } = g sin θ   \frac{1}{1+ \frac{1}{2} }

           a₂ = g sin θ   ⅔

    c) hollow sphere

         I = 2/3 m r²

         a₃ = g sin θ   \frac{1}{1 + \frac{2}{3} }

         a₃ = g sin θ \frac{3}{5}

    d) solid sphere

         I = 2/5 m r²

         a₄ = g sin θ  \frac{1 }{1 + \frac{2}{5} }

         a₄ = g sin θ  \frac{5}{7}

    We already have all the accelerations, to facilitate the comparison let’s place the fractions with the same denominator (the greatest common denominator is 210)

    a) a₁ = g sin θ ½ = g sin θ      \frac{105}{210}

    b) a₂ = g sinθ ⅔ = g sin θ     \frac{140}{210}

    c) a₃ = g sin θ \frac{3}{5}= g sin θ       \frac{126}{210}

    d) a₄ = g sin θ \frac{5}{7} = g sin θ      \frac{150}{210}

    the order of acceleration from lower to higher is

       

         a₁ <a₃ <a₂ <a₄

    acceleration are

         hollow cylinder < hollow sphere < solid cylinder < solid sphere

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