A 4.0 kg wheel of 20 cm radius of gyration is rotating at 360 rpm. The retarding frictional torque is .12 N.m. Compute the time it will take the wheel to coast to a rest.
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To solve this problem, we need to use the equation of motion for rotational motion:
Torque = Moment of Inertia * Angular Acceleration
Given:
Mass (m) = 4.0 kg
Radius of gyration (k) = 20 cm = 0.2 m
Rotational speed (ω) = 360 rpm
First, we need to convert the rotational speed from rpm to rad/s:
ω = 360 rpm * (2π rad/1 min) * (1 min/60 s)
ω = 2π * 360/60 rad/s
ω = 12π rad/s
Next, we need to calculate the moment of inertia (I) using the formula:
I = mk^2
I = 4.0 kg * (0.2 m)^2
I = 0.16 kg.m^2
Since the retarding frictional torque (τ) is given, we can rearrange the equation above to solve for angular acceleration (α):
τ = I * α
α = τ / I
α = 0.12 N.m / 0.16 kg.m^2
α = 0.75 N.m / kg.m^2
Now, we can use the relation between angular acceleration and final angular velocity to find the time (t):
ω = ω0 + αt
0 = 12π rad/s + (-0.75 N.m / kg.m^2) * t
-12π = -0.75π t
Simplifying the equation, we get:
0.75π t = 12π
t = 12π / (0.75π)
t = 16 seconds
Therefore, it will take 16 seconds for the wheel to coast to a rest.
To compute the time it will take the wheel to coast to a rest, we need to use the concepts of angular acceleration, angular velocity, and torque. Here's how you can calculate it:
1. Convert the angular velocity from rpm to rad/s:
- Angular velocity (ω) = 360 rpm * (2π rad/1 min) * (1 min/60 s) = 37.7 rad/s
2. The torque equation is given by Torque (τ) = Moment of inertia (I) * Angular acceleration (α). Rearranging the equation gives:
- Angular acceleration (α) = Torque (τ) / Moment of inertia (I)
3. The moment of inertia of a wheel can be calculated using the formula:
- Moment of inertia (I) = 0.5 * m * r^2
where m is the mass of the wheel and r is the radius of gyration.
- Substitute the values:
- m = 4.0 kg
- r = 0.20 m (since the radius of gyration is given as 20 cm)
- I = 0.5 * 4.0 kg * (0.20 m)^2 = 0.16 kg.m^2
4. Substitute the given values into the torque equation:
- α = τ / I
- α = 0.12 N.m / 0.16 kg.m^2 = 0.75 rad/s^2
5. The final step is to use the equation for angular acceleration to calculate the time taken for the wheel to coast to a rest:
- Angular acceleration (α) = (Change in angular velocity) / Time
- We know that the final angular velocity is zero when the wheel comes to rest, so the change in angular velocity is -ω (negative of the initial angular velocity) and the time is what we're trying to find.
- Rearranging the equation:
- Time = -ω / α
- Time = -37.7 rad/s / 0.75 rad/s^2 = -50.27 s
However, note that the negative sign indicates direction, so we take the absolute value of the time:
- Time = 50.27 s
Therefore, the wheel will take approximately 50.27 seconds to coast to a rest.