Answer:
τ = 1679.68Nm
Explanation:
In order to calculate the required torque you first take into account the following formula:
[tex]\tau=I\alpha[/tex] (1)
τ: torque
I: moment of inertia of the merry-go-round
α: angular acceleration
Next, you use the following formulas for the calculation of the angular acceleration and the moment of inertia:
[tex]\omega=\omega_o+\alpha t[/tex] (2)
[tex]I=\frac{1}{2}MR^2[/tex] (3) (it is considered that the merry-go-round is a disk)
w: final angular speed = 3.1 rad/s
wo: initial angular speed = 0 rad/s
M: mass of the merry-go-round = 432 kg
R: radius of the merry-go-round = 2.3m
You solve the equation (2) for α. Furthermore you calculate the moment of inertia:
[tex]\alpha=\frac{\omega}{t}=\frac{3.1rad/s}{2.1s}=1.47\frac{rad}{s^2}\\\\I=\frac{1}{2}(432kg)(2.3)^2=1142.64kg\frac{m}{s}[/tex]
Finally, you replace the values of the moment of inertia and angular acceleration in the equation (1):
[tex]\tau=(1142.64kgm/s)(1.47rad/s^2)=1679.68Nm[/tex]
The required torque is 1679.68Nm
