Question

The objects listed are placed at the top of a ramp and roll down to the bottom without slipping. Assuming that there is no air resistance, rank them in order from fastest average rolling speed to slowest. a. large marble b. basteball c. manhole cover d. wedding ring

Answer 1

Explanation:

For each object, the initial potential energy is converted to rotational energy and translational energy:

PE = RE + KE

mgh = ½ Iω² + ½ mv²

For the marble (a solid sphere), I = ⅖ mr².

For the basketball (a hollow sphere), I = ⅔ mr².

For the manhole cover (a solid cylinder), I = ½ mr².

For the wedding ring (a hollow cylinder), I = mr².

If we say k is the coefficient in each case:

mgh = ½ (kmr²) ω² + ½ mv²

For rolling without slipping, ωr = v:

mgh = ½ kmv² + ½ mv²

gh = ½ kv² + ½ v²

2gh = (k + 1) v²

v² = 2gh / (k + 1)

The smaller the value of k, the higher the velocity.  Therefore:

marble > manhole cover > basketball > wedding ring

Answer 2

The arrangement of the given objects in order of their increasing speed down the ramp is wedding ring > large marble  >  manhole cover  > basketball.

Apply the principle of conservation of energy;

$\frac{1}{2} mv^2 \ + \ \frac{1}{2} I \omega ^2 = mgh$

where;

• I is moment of inertia

The moment of inertia for the marble (a solid sphere);

$I = \frac{2}{5} mr^2$

The moment of inertia for the basketball (a hollow sphere);

$I = \frac{2}{3} mr^2$

The moment of inertia for the manhole cover (a solid cylinder);

$I = \frac{1}{2} mr^2$

The moment of inertia for the wedding ring (a hollow cylinder);

$I = mr^2$

$\frac{1}{2} mv^2 \ + \ \frac{1}{2} I (\frac{v}{r} )^2 = mgh\\\\\frac{1}{2} mv^2 \ + \ \frac{1}{2r^2} Iv^2 = mgh\\\\v^2 (m + \frac{I}{r^2} )= 2mgh\\\\v^2 = \frac{2mgh}{m + \frac{I}{r^2} } \\\\v = \sqrt{\frac{2mgh}{m + \frac{I}{r^2} } }$

The smaller the value of moment of inertia, the greater the value the speed of the object.

The arrangement of the given objects in order of their increasing speed down the ramp;

wedding ring > large marble  >  manhole cover  > basketball

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