How long would a braking torque of 3.17 N·m have to act to just stop a spinning wheel that has an initial angular momentum of 6.67 kg·m2/s?
To solve this problem, we can use the principle of conservation of angular momentum. Initially, the spinning wheel has an angular momentum of 6.67 kg·m²/s. The braking torque applied to the wheel is 3.17 N·m.
The equation for angular momentum (L) is given by:
L = Iω
Where:
L is the angular momentum,
I is the moment of inertia, and
ω is the angular velocity.
We are given the initial angular momentum (L), so we can rearrange the equation to solve for the initial angular velocity (ω):
ω = L / I
Given that the angular momentum (L) is 6.67 kg·m²/s and the moment of inertia (I) is not provided, we need to find I to calculate ω.
The moment of inertia of a spinning object can be calculated using the equation:
I = m × r²
Where:
I is the moment of inertia,
m is the mass of the object, and
r is the radius of rotation.
Since the mass of the wheel is not provided, we need to find it. However, the mass is not given directly. Instead, we know the moment of inertia (I) and the radius of rotation (r). Rearranging the formula for moment of inertia, we get:
m = I / r²
Given that the moment of inertia (I) is 6.67 kg·m²/s and the radius of rotation (r) is not provided, we need to find r to calculate m.
Now, we can rearrange the formula for radius to solve for r:
r = sqrt(I / m)
Substituting the given values for I and m, we can find the radius (r) of the spinning wheel.
Once we have the radius (r), we can substitute the values of I and ω into the equation ω = L / I to calculate the initial angular velocity (ω).
Finally, to find the time (t) required to stop the spinning wheel using the braking torque (τ), we can use the equation:
τ = Iα
Where:
τ is the braking torque, and
α is the angular acceleration.
The angular acceleration can be calculated by rearranging the equation:
α = ω / t
Rearranging this equation, we get:
t = ω / α
Substituting the known values of ω and τ, we can determine the time (t) required for the braking torque to stop the spinning wheel.