Rotational Motion - I have the answer but I don't know how to get there

[jbcaviness]

A torque of 1 N.m is applied to a bike wheel of radius 35 cm and mass 0.75 kg. Treating the wheel as a hoop (I=MR^2)

A. What is it's angular acceleration? Answer - 10.9 rad/s^2
B. If the wheel starts from rest, what is its angular speed after 5 seconds? Answer - 54.4 rad/s
C. How many turns does it do in those 5 seconds? Answer 21.7 turns

I'm trying to study for my exam, and I'm stuck on this problem for our practice test. My professor gave us answers, but I'm just at a loss as to how to get the answers. Can anyone explain how to solve this problem? Thanks!

 

[Fightfish]

What is the relationship between torque, moment of inertia and angular acceleration?

 

[kerimek]

To solve this problem, you need to use the equation for rotational motion: torque = moment of inertia x angular acceleration (τ = Iα). In this case, the moment of inertia (I) for a hoop is equal to the mass (M) multiplied by the radius squared (R^2). So, we can rewrite the equation as τ = MR^2α.

Now, we know that the torque (τ) is 1 N.m and the mass (M) is 0.75 kg. The radius (R) is given as 35 cm, but we need to convert it to meters by dividing by 100. So, R = 0.35 m. Plugging these values into the equation, we get:

1 N.m = (0.75 kg)(0.35 m)^2α

Solving for α, we get α = 10.9 rad/s^2. This is the answer to part A.

For part B, we can use the equation ω = ω0 + αt, where ω0 is the initial angular speed (which is 0 since the wheel starts from rest), α is the angular acceleration we just calculated, and t is the time (5 seconds). So, ω = 0 + (10.9 rad/s^2)(5 s) = 54.4 rad/s.

For part C, we can use the equation θ = θ0 + ω0t + 1/2αt^2, where θ is the angular displacement, θ0 is the initial angular displacement (which is 0 since the wheel starts from rest), ω0 is the initial angular speed (also 0), α is the angular acceleration we calculated, and t is the time (5 seconds). Since we are looking for the number of turns, we need to convert the angular displacement from radians to turns by dividing by 2π. So, θ = (0) + (0)(5) + 1/2(10.9 rad/s^2)(5 s)^2 = 21.7 turns.

I hope this explanation helps you understand how to solve this problem. Remember to always use the correct equations and units when solving rotational motion problems. Good luck on your exam!