The angular acceleration a of a spinning top as a function of t is : alpha = 3t^(2)+ 5t. At t = 0, the angular velocity omega_(0) = 10 rad/s and angular position theta = 8 rad. The angular position as a function of time t is given by which of the following expression?

The angular acceleration a of a spinning top as a function of t is : a

1.	The angular acceleration a of a spinning top as a function of t is : alpha = 3t^(2)+ 5t. At t = 0, the angular velocity omega_(0) = 10 rad/s and angular position theta = 8 rad. The angular position as a function of time t is given by which of the following expression?
Answer» `(1)/(4) t^(4) + (5)/(6) t^(3) + 10 t + 8` `(5)/(6) t^(4) + (1)/(4) t^(3) + (2)/(5) t + 8` `2t^(4) + 3t^(3) + 5t + 8` `(1)/(4) t^(4) + (3)/(4) t^(3) + 6t^(2) + 8` Solution :Given : `ALPHA = 3t^(2) + 5t` `alpha = (d^(2) THETA)/(dt^(2)) = 3t^(2) + 5t "" … (i)` `alpha = (d^(2) theta)/(dt^(2)) = 3t^(2) +5t "" …. (i)` Integrate both sides w.r.t. t in equation (i) , we get `(d theta)/(dt) = t^(3) + (5)/(2) t^(2) + C_(1)` where `C_(1)` is a constant of integration . or `omega = (d theta)/(dt) = t^(3) + (5)/(2) t^(2) + C_(1)` Using initial conditions , at t = 0 , `omega_(0) = 10` RAD/s `THEREFORE C_(1) = 10` rad/s or `(d theta)/(dt) = t^(3) + (5)/(2) t^(2) + 10 "" .... (ii)` Integrate both sides w.r.t t in equation (ii) , we get `theta = (1)/(4) t^(4) + (5)/(6) t^(3) + 10 t + C_(2)` where `C_(2)` is a constant of integration . Using initial conditions , at t = 0 , `theta_(0) = 8` rad `therefore C_(2) = 8` rad or `theta = (1)/(4) t^(4) + (5)/(6) t^(3) + 10 t + 8`