Question 356: Centre of Mass and Rotational Motion

Question

A pulley of radius $$2$$ $$m$$ is rotated about its axis by a force $$F = \left( {20t - 5{t^2}} \right)$$ newton (where $$t$$ is measured in seconds)  applied tangentially. If the moment of inertia of the pulley about its axis of rotation is $$10kg$$-$${m^2}$$ the number of rotation made by the pulley before its direction of motion is reversed, is:

Accepted Answer

Given $$F = 20t - 5{t^2}$$, R = 2 m and $$I$$ = 10 kg m
2
Torque applied on pulley  $$	au  = FR$$
$$	herefore$$ $$\alpha  = {{FR} \over I}$$ [ as $$	au  = I\alpha $$ ]
$$ \Rightarrow $$ $$\alpha  = {{\left( {20t - 5{t^2}} ight) 	imes 2} \over {10}}$$
$$ \Rightarrow $$ $$\alpha  = 4t - {t^2}$$
$$ \Rightarrow {{d\omega } \over {dt}} = 4t - {t^2}$$
$$ \Rightarrow \int\limits_0^\omega  {d\omega }  = \int\limits_0^t {\left( {4t - {t^2}} ight)} dt$$
$$ \Rightarrow \omega  = 2{t^2} - {{{r^3}} \over 3}$$
( At $$t = 0,6 \,s$$ $$\omega  = 0$$  )
$$\omega  = {{d	heta } \over {dt}} = 2{t^2} - {{{t^3}} \over 3}$$
$$\int\limits_0^	heta  {d	heta }  = \int\limits_0^6 {\left( {2{t^2} - {{{r^3}} \over 3}} ight)} dt$$
$$ \Rightarrow 	heta  = 36rad\,\, \Rightarrow n = {{36} \over {2\pi }} < 6$$

Answer

more than $$3$$ but less than $$6$$

Answer

more than $$6$$ but less than $$9$$

Answer

more than $$9$$

Answer

less than $$3$$

Centre of Mass and Rotational Motion

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