To determine which hoop has the greater angular acceleration and how many times greater it is compared to the other hoop, we need to analyze the situation using the concepts of torque, moment of inertia, and angular acceleration.
#### Solution By Steps
***Step 1: Identify the Moment of Inertia for Each Hoop***
The moment of inertia (I) for a hoop is given by the formula:
I = M * r^2
For the smaller hoop (radius = R):
I_small = M * R^2
For the larger hoop (radius = 2R):
I_large = M * (2R)^2 = M * 4R^2
***Step 2: Calculate the Torque for Each Hoop***
Torque (τ) is given by the formula:
τ = F * r
For the smaller hoop:
τ_small = F * R
For the larger hoop:
τ_large = F * (2R) = 2F * R
***Step 3: Relate Torque to Angular Acceleration***
Using the relationship between torque, moment of inertia, and angular acceleration (α):
τ = I * α
For the smaller hoop:
F * R = (M * R^2) * α_small
=> α_small = (F * R) / (M * R^2) = F / (M * R)
For the larger hoop:
2F * R = (M * 4R^2) * α_large
=> α_large = (2F * R) / (M * 4R^2) = (2F) / (M * 4R) = F / (2M * R)
***Step 4: Compare Angular Accelerations***
Now we compare α_small and α_large:
α_small = F / (M * R)
α_large = F / (2M * R)
To find how many times greater α_small is than α_large:
α_small / α_large = (F / (M * R)) / (F / (2M * R)) = 2
Thus, the smaller hoop has an angular acceleration that is 2 times greater than that of the larger hoop.
#### Final Answer
The smaller hoop; two times as great.