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# A solid cylindrical disk whose mass is 1.4 kg and radius is 8.5 cm, rolls across a horizontal table at a speed of 15 cm/...

A solid cylindrical disk whose mass is 1.4 kg and radius is 8.5 cm, rolls across a horizontal table at a speed of 15 cm/s. What is the instantaneous velocity of the top of the rolling cylinder? What is the angular speed of the rolling disk? What is the kinetic energy of the rolling disk?

Given:

The mass of the cylinder is 1.4 kg.

The radius of the cylinder is 8.5 cm.

The speed of the cylinder is 15 cm/s.

Calculation:

Part-I:

Write the expression for the angular speed.

$\omega =\frac{{v}_{cm}}{R}$

Here, $\omega$ is the angular speed, R is the radius, and vcm is the speed of the center of the mass.

Substitute, 15 cm/s for vcm, and 8.5 cm for R in the above expression.

Thus, the angular speed is 1.765 rad/s.

Part-II:

Write the expression for the instantaneous velocity of the top of the cylinder.

$v={v}_{cm}+R\omega$

Here, v is the instantaneous velocity of the top of the cylinder.

Substitute, 1.765 rad/s for $\omega$, 8.5 cm for R, and 15 cm/s for vcm, in the above expression.

Thus, the instantaneous velocity of the top edge of the cylinder is 30 cm/s.

Part-III:

Write the expression for the total kinetic energy of the cylinder.

total kinetic energy = rotational kinetic energy + translational kinetic energy.

${E}_{k}=\frac{3}{4}m{{v}_{cm}}^{2}$

Substitute, 15 cm/s for vcm, and 1.4 kg for m in the above expression.

Thus, the kinetic energy of the cylinder is 0.023525 J.

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