91Ó°ÊÓ

1. The mass of cylinder is,$\mathrm{m}=2.0\text{ k²µ}$
2. ${\mathrm{F}}_1=6.0\text{ N}$
3. ${\mathrm{F}}_2=4.0\text{ N}$
4. ${\mathrm{F}}_3=2.0\text{ N}$
5. $\mathrm{R}=0.12\text{″¾}$
6. $\mathrm{r}=0.050\text{″¾}$

From the given figure and information, it is possible to find the net torque acting on the cylinder. Using this net torque and value of moment of inertia of a cylinder, you can find the magnitude of the angular acceleration of the cylinder. The formulas used for above are given below.

1. Net Torque, ${\mathbf{τ}}_{\mathbf{net}}\mathbf{=}{\mathbf{F}}_{\mathbf{1}}\mathbf{R}\mathbf{−}{\mathbf{F}}_{\mathbf{2}}\mathbf{R}\mathbf{−}{\mathbf{F}}_{\mathbf{3}}\mathbf{r}$
2. Angular acceleration,$\mathbf{Î\pm }\mathbf{=}{\mathbf{Ï„}}_{\mathbf{net}}\mathbf{/}\mathbf{I}$
3. Moment of inertia,$\mathbf{I}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{}{\mathbf{MR}}^{\mathbf{2}}$

Calculating net torque by the above formula,

${\mathrm{τ}}_{\mathrm{net}}={\mathrm{F}}_1\mathrm{R}−{\mathrm{F}}_2\mathrm{R}−{\mathrm{F}}_3\mathrm{r}$

Substitute all the value in the above equation

$\begin{array}{c}{\mathrm{Ï„}}_{\mathrm{net}}=\left(6.0\text{ N}\right)\left(0.12\text{″¾}\right)−\left(4.0\text{ N}\right)\left(0.12\text{″¾}\right)−\left(2.0\text{ N}\right)\left(0.050\text{″¾}\right)\\ =71\text{ N³¾}\end{array}$

Also, Moment of inertia will be

$\begin{array}{c}\mathrm{I}=\frac{1}{2}{\mathrm{MR}}^2\\ =\frac{1}{2}\left(2.0\text{ k²µ}\right)×{\left(0.12\text{″¾}\right)}^2\\ =0.0144{\text{ k²µ.m}}^{\text{2}}\end{array}$

Now, substituting these values to find the angular acceleration of the cylinder

We have,

$\mathrm{Î\pm }=\frac{{\mathrm{Ï„}}_{\mathrm{net}}}{\mathrm{I}}$

Substitute all the value in the above equation

$\begin{array}{c}\mathrm{Î\pm }=\frac{71\text{ N³¾}}{0.014{\text{ k²µ.m}}^{\text{2}}}\\ =9.7\text{ r²¹»å}/{\text{s}}^{\text{2}}\end{array}$

Hence the angular acceleration is, $9.7\text{ r²¹»å}/{\text{s}}^{\text{2}}$.

Since the value of acceleration is position, the direction of the angular acceleration of the cylinder is, counter clockwise.