When a 19.9 kg wheel with an angular speed of 3.10 rad/s is disconnected from a motor, a 0.124 N · m frictional torque slows the wheel to a stop. If the wheel has radius of 0.71 m, how long will it take for the wheel to come to rest after being disconnected from the motor?
To calculate the time it takes for the wheel to come to rest after being disconnected from the motor, we can use the equation for angular acceleration involving torque.
The equation is:
τ = I α
Where:
τ is the torque applied to the object,
I is the moment of inertia of the object,
α is the angular acceleration of the object.
We can rearrange the equation to solve for α:
α = τ / I
Next, we need to find the moment of inertia of the wheel. The moment of inertia of a solid disk rotating about its center is given by the equation:
I = (1/2) m r^2
Where:
m is the mass of the wheel,
r is the radius of the wheel.
Now, let's substitute the given values into the equations:
Given:
m = 19.9 kg
r = 0.71 m
τ = 0.124 N·m
First, calculate the moment of inertia:
I = (1/2) m r^2
I = (1/2) * (19.9 kg) * (0.71 m)^2
Next, calculate the angular acceleration:
α = τ / I
Now that we have the angular acceleration, we can use it to find the time it takes for the wheel to come to rest.
The formula linking angular speed, angular acceleration, and time is:
ω = ω₀ + αt
Where:
ω is the final angular speed (0 rad/s),
ω₀ is the initial angular speed (3.10 rad/s),
α is the angular acceleration,
t is the time taken.
Rearranging the equation, we can solve for t:
t = (ω - ω₀) / α
Substituting the known values:
t = (0 rad/s - 3.10 rad/s) / α
Calculate the value of α from the given torque and moment of inertia, and then substitute it into the equation to find t.