91Ó°ÊÓ

1. Radius of pulley A is$r_a=15\text{ c³¾}$
2. Radius of pulley B is$r_b=10\text{ }\text{cm}$
3. Radius of pulley B’ is$r_{b^{\text{'}}}=5\text{ c³¾}$
4. Radius of pulley C is$r_c=25\text{ }\text{cm}$
5. Angular velocity of pulley A is${\mathrm{Ó\neg }}_a=10\text{ r²¹»å/²õ}$
6. If the belt between two pulleys does not slip, the linear speed at the rims of the two pulleys must be equal.

The linear velocity is given as the rate of change of displacement with respect to time. The angular velocity is defined as the rate of change of angular displacement with respect to time. Find the linearvelocity of belt 1 from radius of pulley A and angular speed of pulley A. From linear speed of belt 1, find the remaining values.

The relation between linear and angular velocity is-

$\mathbf{v}\mathbf{=}\mathbf{rÓ\neg }$

where, v is velocity, ris radius and$\mathbf{Ó\neg }$is angular velocity.

Use the following formula to find linear speed,

$\begin{array}{c}{v}_a=r_a{\mathrm{Ó\neg }}_a\\ =0.15\text{″¾}×10\text{ r²¹»å}/\text{s}\\ =1.5\text{ }\text{m/s}\end{array}$

Hence,linear speed of a point on belt$1$is$1.5\text{″¾/²õ}$.

Linear speed of pulley B is also$1.5\text{″¾}/\text{s}$because the belt doesn’t slip.

So, angular speed is given as follows:

$\begin{array}{c}{\mathrm{Ó\neg }}_b=\frac{v_b}{r_b}\\ =\frac{1.5\text{″¾}/\text{s}}{0.10\text{″¾}}\\ =15\text{ r²¹»å/²õ}\end{array}$

Hence,angular speed of pulley$\text{B}$is$15\text{ r²¹»å/²õ}$.

Angular speed of pulley$\text{B'}$is also$15\text{ }\text{rad/s}$because it is concentric to pulley A.

Hence, angular speed of pulley B’ is$15\text{ }\text{rad/s}$ .

Linear speed is given as follows:

$\begin{array}{c}{v}_2=r_{b^{\text{'}}}×{\mathrm{Ó\neg }}_{b^{\text{'}}}\\ =0.05\text{″¾}×15\text{ r²¹»å}/\text{s}\\ =0.75\text{″¾/²õ}\end{array}$

Hence, linear speed of point on belt 2 is $0.75\text{ }\text{m/s}$.

Angular speed is as follows:

$\begin{array}{c}{\mathrm{Ó\neg }}_c=\frac{v_2}{r_c}\\ =\frac{0.75\text{″¾}/\text{s}}{0.25\text{″¾}}\\ =3\text{rad/s}\end{array}$

Hence, angular speed of pulley C is $3\text{ r²¹»å/²õ}$$3\text{ r²¹»å/²õ}$.