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anonymous

  • 5 years ago

\[\large\mathsf{\text{Problem Based on Rotational Motion}}\]

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  1. anonymous
    • 5 years ago
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    A spool of mass \(\mathsf{m}\) and inner radius \(\mathsf{r}\) and outer radius \(\mathsf{2r}\), having moment of inertia \(\Large\mathsf{\frac{mr^2}{2}}\) is made to roll without sliding on a rough horizontal surface by the help of an applied force \(\mathsf {(F = mg)}\), on ideal string wrapped around the inner cylinder (Shown in the figure). |dw:1326937711360:dw| Find the minimum Co-efficient of Friction required for Pure Rolling.

  2. anonymous
    • 5 years ago
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    Here's what I did. |dw:1326938034465:dw| For Pure Rolling, \(\mathsf{\omega . r = v \tag{1}}\) \(\mathsf{\alpha . r = a }\tag{2}\) Two torques are acting on the Spool about it's center (Circle's center), one due to the applied force \(\mathsf F\) and other due to the friction generated by the rolling of the spool on the rough horizontal surface. \[\mathsf{\sum \tau = 2r. f_{k} - F.r = I \alpha }\tag 3\] Translation motion's equation, \[\mathsf{ F - f_{k} = Ma} \tag 4\] Solving equations (2), (3) and (4) and using \(\Large\mathsf{ \mu_{k} = \frac{f_{k}}{N}}\). I am getting \(\Large\mu = \frac{3}{5}\). While the options are \(\large\mathsf{\frac{2}{9}, \frac{4}{9}, \frac{5}{9} \text{and 'none of these'}}\).

  3. anonymous
    • 5 years ago
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    Double check a few things. First, note that\[\alpha (2r) = a_{CM}\]Also, note that\[\sum \vec {\bf F}_x = 0\]Otherwise, \[\alpha (2r) = a_{CM}\]won't be satisfied.

  4. anonymous
    • 5 years ago
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    Oh yeah, Thank You so much! I knew I did something silly.

  5. anonymous
    • 5 years ago
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    Did you get the correct answer?

  6. anonymous
    • 5 years ago
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    Yeah, I think so \(\Large\mathsf{\mu_{k} =\frac{5}{9}}\). Thanks!

  7. anonymous
    • 5 years ago
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    Excellent.

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