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A uniform marble rolls down a symmetrical bowl, start- ing from rest at the top of the left side. The top of each side is a distance h above the bottom of the bowl. The left half of the bowl is rough enough to cause the marble to roll without slipping, but the right half has no friction because it is coated with oil. (a) How far up the smooth side will the marble go, measured vertically from the bottom? (b) How high would the marble go if both sides were as rough as the left side? (c) How do you account for the fact that the marble goes higher with friction on the right side than without friction?

Respuesta :

Answer:

Part a)

[tex]h' = \frac{10}{14} h[/tex]

Part b)

if both sides are rough then it will reach the same height on the other side because the energy is being conserved.

Part c)

Since marble will go to same height when it is rough while when it is smooth then it will go to the height

[tex]h' = \frac{10}{14} h[/tex]

so on smooth it will go to lower height

Explanation:

As we know by energy conservation the total energy at the bottom of the bowl is given as

[tex]\frac{1}{2} mv^2 + \frac{1}{2}I\omega^2 = mgh[/tex]

here we know that on the left side the ball is rolling due to which it is having rotational and transnational both kinetic energy

now on the right side of the bowl there is no friction

so its rotational kinetic energy will not change and remains the same

so it will have

[tex]\frac{1}{2}mv^2 = mgh'[/tex]

now we know that

[tex]I = \frac{2}{5}mr^2[/tex]

[tex]\omega = \frac{v}{r}[/tex]

so we have

[tex]\frac{1}{2}mv^2 + \frac{1}{2}(\frac{2}{5}mr^2)(\frac{v}{r})^2 = mgh[/tex]

[tex]\frac{1}{2}mv^2 + \frac{1}{5}mv^2 = mgh[/tex]

[tex]\frac{7}{10}mv^2 = mgh[/tex]

[tex]\frac{1}{2}mv^2 = \frac{10}{14}mgh[/tex]

so the height on the smooth side is given as

[tex]h' = \frac{10}{14} h[/tex]

Part b)

if both sides are rough then it will reach the same height on the other side because the energy is being conserved.

Part c)

Since marble will go to same height when it is rough while when it is smooth then it will go to the height

[tex]h' = \frac{10}{14} h[/tex]

so on smooth it will go to lower height

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