$\"\"$

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Step 1:

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The hoop rolling on a horizontal surface at speed is $\"\"$ .

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The hoop inclined at an angle is $\"\"$ .

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The potential energy of the hoop at inclination is $\"\"$ .

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The kinetic energy of the hoop is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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The kinetic energy of the hoop is $\"\"$ .

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Law of conservation of energy: $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The hoop inclined up is $\"\"$ .

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(b)

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How long will it be on the incline before it arrives back at the bottom

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The hoop inclined up is $\"\"$ .

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The time taken to reach the top is $\"\"$ .

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$\"\"$

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$ .

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The total time taken for the hoop to incline before it arrives back at the bottom is $\"\"$ .

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The total time taken for the hoop to incline before it arrives back at the bottom is $\"\"$ . \ \