Question

A solid sphere of mass 4.0kg and radius 0.12m is at rest at the top of a ramp incline 15 degrees. It rolls to th bottom without sipping.The upper end of the ramp is 1.2m higher than the lower end, find the sphere's total kinetic energy when it reaches the bottom

Answer 1

The loss in gravitational potential energy is equal to the gain in translational kinetic energy and rotational kinetic energy for the case of maximum velocity of sphere when it reaches ground.

⟹mgh=

2

1

mv

2

+

2

1

Iω

2

⟹mgh=

2

1

mv

2

+

2

1

(

5

2

mR

2

)ω

2

From the no-slip condition:

v=Rω

⟹mgh=

2

1

mv

2

+

10

2

mv

2

⟹v=

7

10gh

$The loss in gravitational potential energy is equal to the gain in translational kinetic energy and rotational kinetic energy for the case of maximum velocity of sphere when it reaches ground.</p><p></p><p>⟹mgh=21mv2+21Iω2</p><p></p><p></p><p>⟹mgh=21mv2+21(52mR2)ω2</p><p></p><p></p><p>From the no-slip condition:</p><p></p><p>v=Rω</p><p></p><p>⟹mgh=21mv2+102mv2</p><p></p><p>⟹v=710gh</p><p></p><p></p><p>$

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