A cylinder has inner and outer radii of $8 c m$ $8 cm$ and $16 c m$ $16 cm$ , respectively, and a mass of $6 k g$ $6 kg$ . If the cylinder's frequency of counterclockwise rotation about its center changes from $2 H z$ $2 Hz$ to $7 H z$ $7 Hz$ , by how much does its angular momentum change?

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A cylinder has inner and outer radii of $8 c m$ $8 cm$ and $16 c m$ $16 cm$ , respectively, and a mass of $6 k g$ $6 kg$ . If the cylinder's frequency of counterclockwise rotation about its center changes from $2 H z$ $2 Hz$ to $7 H z$ $7 Hz$ , by how much does its angular momentum change?

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Answer 1 · 2016-12-18T18:18:36

The nasty part of this problem is in coming up with the moment of inertia for a thick-walled cylinder rotating along an axis through its centre. This is

$I = \frac{m}{2} (r_{1}^{2} + r_{2}^{2})$ $I= m/2(r_1^2 +r_2^2)$

where $r_{1}$ $r_1$ and $r_{2}$ $r_2$ are the inner and outer radii, respectively.

In this case, $I = \frac{6}{2} ({.08}^{2} + {.16}^{2})$ $I= 6/2(.08^2 +.16^2)$ = $0.096 k g m^{2}$ $0.096 kgm^2$

Now, the angular momentum is defined as the product of its moment of inertia multiplied by the angular momentum (in radians/s don't forget!).

$L = I$ $L=I$ $ω$ $omega$

Since 1 Hz is equivalent to 2 $π$ $pi$ radians, the angular velocity changes from 4 $π$ $pi$ to 14 $π$ $pi$ , and so, the change in angular momentum is found as follows:

$L_{i} = 0.096 \times 4 π$ $L_i = 0.096 xx 4"pi$ = $0.384$ $0.384$ pi#

$L_{f} = 0.096 \times 14 π$ $L_f = 0.096 xx 14"pi$ = $1.344$ $1.344$ pi#

$DeltaL = 0.960pi$ $kgm^2/s$

(Hope you don't mind that I held back on the units until the last line. I thought they would make things a bit messy!)

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A cylinder has inner and outer radii of $8 c m$ $8 cm$ and $16 c m$ $16 cm$ , respectively, and a mass of $6 k g$ $6 kg$ . If the cylinder's frequency of counterclockwise rotation about its center changes from $2 H z$ $2 Hz$ to $7 H z$ $7 Hz$ , by how much does its angular momentum change?

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A cylinder has inner and outer radii of $8 c m$ $8 cm$ and $16 c m$ $16 cm$ , respectively, and a mass of $6 k g$ $6 kg$ . If the cylinder's frequency of counterclockwise rotation about its center changes from $2 H z$ $2 Hz$ to $7 H z$ $7 Hz$ , by how much does its angular momentum change?