A belt moves over two pulleys A and B as shown in the figure. The pulleys are mounted on two fixed horizontal axles. Radii of the pulleys A and B are 50 cm and 80 cm respectively. Pulley A is driven at constant angular acceleration of 0.8 `"rad/s"^2` unitl pulley B acquries an angular velocity of 10 rad/s. The belt does not slide on either of the pulleys.

(a)Find the acceleration of a point C on the belt and angular acceleration of the pulley B.
(b) How long does it take for the pulley B to acquire an angular velocity of 10 rad/s ?

Explanation

Since the belt does not slide on the pulleys, magnitudes of velocity and acceleration of any point on the belt are same as that of any point on the peridhery of either of the two pulleys.
`veca_(T) = vecalphaxxvecr" "a_c = alpha_c = alpha_A r_A = alpha_B r_B`
Substituting `r_A = 0.5 m, r_B =0.8` m and ` a_A = 0.8 "rad/s"^2`
we have ` a_c = 0.4 m//s^2 ` and
`alpha_B = a_c/r_B = (alpha_A r_B)/r_B = 0.5 rad//s^2`
we have `a_c = 0.4 m//s^2` and `alpha_B = a_c/r_B = (alpha_A r_A)/r_B = 0.5 rad//s^2`
`because omega + omega_0 alpha t rArr t =(omega_B - omega_(B_(0)))/alpha_B`
Substituting `W_(Bo) 0, omega_B =10` rad/s and
` alpha_B = 0.5 "rad/s"^2,` we have t = 20 s