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Q7.30     A solid disc and a ring, both of radius 10cm  are placed on a horizontal table
              simultaneously, with initial angular speed equal to 10\Pi rad\: s^{-1} . Which of the two will start to roll
             earlier ? The co-efficient of kinetic friction is \mu _{k}=0.2.

Answers (1)

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Friction is the cause for motion here.

So using Newton's law of motion we can write :

                                             f\ =\ ma

                                        \mu _k mg\ =\ ma

or                                           a\ =\ \mu _k g

Now by the equation of motion, we can write :

                                              v\ =\ u\ +\ at

or                                            v\ =\ \mu _k gt

The torque is given by :

                                             \tau\ =\ I \alpha

or                                  r\times f \ =\ -I \alpha

or                                   \mu _k mgr \ =\ -I \alpha

or                                             \alpha\ =\ -\frac{\mu _k mgr}{I}

Now using the equation of rotational motion we can write :

                                          \omega\ =\ \omega _o\ +\ \alpha t

or                                      \omega\ =\ \omega _o \ +\ -\frac{\mu _k mgr}{I}t

Condition for rolling is         v\ =\ \omega r

So we can write :

                                           v =\ r\left ( \omega _o \ +\ -\frac{\mu _k mgr}{I}t \right )

For ring the moment of inertia is :   mr^2

So we have : 

                                          t\ =\ \frac{r \omega _o}{2 \mu _k g} 

or                                            =\ \frac{0.1\times 10\times 3.14}{2\times 0.2 \times 10}\ =\ 0.80\ s

 

Now in case of the disc, the moment of inertia is :

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

Thus                                t\ =\ \frac{r \omega _o}{3 \mu _k g}

or                                          =\ \frac{0.1\times 10\times 3.14}{3 \times 0.2 \times 9.8}\ =\ 0.53\ s

Hence disc will start rolling first. 

Posted by

Devendra Khairwa

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