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.
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.
To compute the time it will take for the wheel to coast to a rest, we need to determine the initial angular velocity (ω₀) of the wheel and then use the equations of rotational motion.
Given:
- Mass of the wheel (m) = 4.0 kg
- Radius of gyration (r) = 20 cm = 0.2 m
- Angular velocity (ω) = 360 rpm = (360/60) rad/s = 6π rad/s
- Retarding frictional torque (τ) = 0.12 N⋅m
Step 1: Calculate the initial angular velocity (ω₀)
The wheel is rotating with an angular velocity of 6π rad/s. To find the initial angular velocity, we need to consider the torque equation:
τ = Iα
Where:
- τ is the torque
- I is the moment of inertia
- α is the angular acceleration
In this case, the torque is the retarding frictional torque (τ = -0.12 N⋅m), and the moment of inertia (I) can be calculated using the formula:
I = m⋅r²
Plugging in the given values:
I = 4.0 kg × (0.2 m)²
I = 4.0 kg × 0.04 m²
I = 0.16 kg⋅m²
Now we can rearrange the torque equation to solve for the angular acceleration (α):
α = τ / I
α = -0.12 N⋅m / 0.16 kg⋅m²
α ≈ -0.75 rad/s²
Finally, we can find the initial angular velocity (ω₀) using the equation:
ω - ω₀ = αt
Since the wheel is starting from rest, ω₀ = 0:
6π - 0 = -0.75 × t
Simplifying:
t = (6π) / 0.75
t ≈ 8π seconds
Step 2: Convert the time to minutes and seconds
To express the time in minutes and seconds, we'll divide the total seconds by 60 to get the minutes and take the remaining seconds:
Total seconds = 8π seconds
Total minutes = (8π) / 60 minutes
Remaining seconds = remainder of (8π) / 60
So the time it will take for the wheel to coast to a rest is approximately:
Total minutes: (8π) / 60 minutes
Remaining seconds: remainder of (8π) / 60 seconds
Note: Make sure to round the values to the appropriate decimal places based on the level of accuracy needed.
To solve this problem, we'll need to use the concepts of rotational kinematics and Newton's second law of motion for rotational motion. Here's how you can calculate the time it will take for the wheel to coast to a rest:
1. Determine the moment of inertia (I) of the wheel:
The moment of inertia for a wheel rotating about its center is given by the formula: I = (1/2)mr², where m is the mass of the wheel and r is the radius of gyration.
In this case, the mass of the wheel is given as 4.0 kg and the radius of gyration is 20 cm (0.2 m). Therefore, the moment of inertia is: I = (1/2) * 4.0 kg * (0.2 m)² = 0.16 kg·m².
2. Convert the rotational speed from rpm to rad/s:
The given rotational speed is 360 rpm. To convert this into rad/s, we need to multiply it by (2π/60) since there are 2π radians in one revolution and 60 seconds in one minute.
Therefore, the rotational speed in rad/s is: (360 rpm) * (2π/60) = 37.7 rad/s.
3. Calculate the angular deceleration (α):
The retarding frictional torque (τ) is given as 0.12 N·m. Using Newton's second law of motion for rotation, we can relate torque to angular acceleration: τ = Iα.
Rearranging the equation, we find α = τ/I.
Therefore, the angular deceleration is: α = (0.12 N·m) / (0.16 kg·m²) = 0.75 rad/s².
4. Determine the time it takes to decelerate to zero angular velocity:
The final angular velocity (ωf) when the wheel comes to rest will be zero.
The angular deceleration (α) relates to the change in angular velocity (Δω) and the time (t) through the equation: Δω = αt.
Rearranging the equation, we find t = Δω/α.
In this case, the wheel starts with an angular velocity of 37.7 rad/s and ends at rest (0 rad/s). Therefore, the change in angular velocity is Δω = 0 - 37.7 rad/s = -37.7 rad/s (negative sign indicates deceleration).
Plugging in the values, the time it will take for the wheel to come to rest is: t = (-37.7 rad/s) / (0.75 rad/s²) = -50.3 s. Note that the negative sign indicates that the wheel is decelerating.
To summarize:
The time it will take for the wheel to coast to a rest is approximately 50.3 seconds.