Answer:
The angular accelerations of the hoops are related by the following equation [tex]\alpha _1 = 2\alpha_2[/tex].
Explanation:
Net force on the hoop is given by;
[tex]F_{net} = ma[/tex]
where;
a is linear acceleration
m is the mass
Net torque on the hoop is given by;
[tex]\tau_{net} =I\alpha[/tex]
where;
I is moment of inertia
α is the angular acceleration
But, τ = Fr
[tex]Fr = I \alpha\\\\\alpha = \frac{Fr}{I} \\\\\alpha = \frac{Fr}{mr^2} \\\\\alpha = \frac{F}{mr} \\\\\alpha = \frac{1}{r} (\frac{F}{m} )\\\\(since\ the \ force\ and \ mass \ are \ the \ same, \frac{F}{m} = constant=k)\\\\ \alpha = \frac{k}{r}\\\\k = \alpha r[/tex]
[tex]\alpha _1 r_1= \alpha_2 r_2[/tex]
let the angular acceleration of the smaller hoop = α₁
let the radius of the smaller hoop = r₁
then, the radius of the larger loop, r₂ = 2r₁
let the angular acceleration of the larger hoop = α₂
[tex]\alpha _1 r_1= \alpha_2 r_2\\\\\alpha_2= \frac{ \alpha _1 r_1}{r_2} \\\\\alpha_2=\frac{\alpha _1 r_1}{2r_1} \\\\\alpha_2= \frac{\alpha _1}{2} \\\\\alpha _1 = 2\alpha_2[/tex]
Therefore, the angular accelerations of the hoops are related by the following equation [tex]\alpha _1 = 2\alpha_2[/tex]
