A drum of 60-mm radius is attached to a disk of 120-mm radius. The disk and drum have a total mass of 6 kg and a combined radius of gyration of 90 mm. A cord is attached as shown and pulled with a force P of magnitude 20 N. Knowing that the disk rolls without sliding,

Question

A drum of 60-mm radius is attached to a disk of 120-mm radius. The disk and drum have a total mass of 6 kg and a combined radius of gyration of 90 mm. A cord is attached as shown and pulled with a force P of magnitude 20 N. Knowing that the disk rolls without sliding, determine (a) the angular acceleration of the disk and the acceleration of G, (b) the minimum value of the coefficient of static friction compatible with this motion.

Accepted Answer

[latex]\overline { a } =r\alpha =\left( 0.12m ight) \alpha [/latex]

[latex]\ \overline { I } =m\overline { k } ^{ 2 }[/latex]

[latex]\ =\left( 6kg ight) { \left( 0.09m ight) }^{ 2 }[/latex]

[latex]\ \overline { I } =48.6kg.{ m }^{ 2 }[/latex]

[latex]\ +\uparrow \sum { { M }_{ C } } =\sum { { \left( { M }_{ C } ight) }_{ eff } } :\left( 20N ight) \left( 0.06m ight) =\left( m\overline { a } ight) r\overline { I } \alpha [/latex]

[latex]\ 1.2N.m=\left( 6kg ight) { \left( 0.12m ight) }^{ 2 }\alpha +48.6{ 10 }^{ -3 }kg.{ m }^{ 2 }[/latex]

[latex]\ 1.2=135 imes { 10 }^{ -3 }\alpha[/latex]

(a) [latex] \alpha =8.889rad/{ s }^{ 2 }\quad[/latex]

[latex]\ \alpha =8.89rad/{ s }^{ 2 }\curvearrowright [/latex]

[latex]\ \overline { a } =r\alpha =\left( 0.12m ight) \left( 8.889rad/{ s }^{ 2 } ight)[/latex]

[latex]\ =1.0667m/{ s }^{ 2 }[/latex]

[latex]\ \overline { a } =1.067m/{ s }^{ 2 } ightarrow [/latex]

(b)[latex] \underrightarrow { + } \sum { { F }_{ y } } =\sum { { \left( { F }_{ y } ight) }_{ eff } } :N-mg=0[/latex]

[latex]\ N=\left( 6kg ight) \left( { 9.81s }^{ 2 } ight)[/latex]

[latex]\ N=58.86N\uparrow [/latex]

[latex]\ \underrightarrow { + } \sum { { F }_{ k } } =\sum { { \left( { F }_{ x } ight) }_{ eff } } :20N-F=m\overline { a } [/latex]

[latex]\ 20N-F=\left( 6kg ight) \left( 1.0667m/{ s }^{ 2 } ight)[/latex]

[latex]\ F=13.6N\leftarrow [/latex]

[latex]\ { \left( { \mu }_{ s } ight) }_{ min }=\frac { F }{ N } =\frac { 13.6N }{ 58.86N } [/latex]

[latex]\ { \left( { \mu }_{ s } ight) }_{ min }=0.231[/latex]

Question : A drum of 60-mm radius is attached to a disk of 1...

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