A disk in your computer spins at 5400 rpm. If this disk turns through 12 revolutions before coming to stop, find the disks angular velocity acceleration. Also find the time it takes to stop.
To find the disk's angular velocity acceleration, we need to use the formula:
Angular velocity acceleration (α) = (final angular velocity - initial angular velocity) / time
First, let's find the initial angular velocity.
Given:
The disk spins at 5400 rpm (revolutions per minute).
The disk turns through 12 revolutions before coming to a stop.
To find the initial angular velocity, we need to convert the given rpm to radians per second.
1 revolution = 2π radians
Initial angular velocity = (5400 rpm) × (2π radians/minute) × (1 minute/60 seconds)
= 5400 × 2π / 60 radians/second
≈ 565.48 radians/second (rounded to two decimal places)
Now, let's find the final angular velocity.
Final angular velocity = 0 (because the disk comes to a stop)
Next, let's find the time it takes to stop.
The time it takes to stop can be calculated using the formula:
Time = (Delta angle) / (angular velocity)
Given:
Delta angle = 12 revolutions = 12 × 2π radians
Initial angular velocity = 565.48 radians/second
Time = (12 × 2π radians) / (565.48 radians/second)
= (24π radians) / (565.48 radians/second)
≈ 0.134 seconds (rounded to three decimal places)
Now, let's find the angular velocity acceleration.
Angular velocity acceleration (α) = (final angular velocity - initial angular velocity) / time
Using:
Final angular velocity = 0 radians/second
Initial angular velocity = 565.48 radians/second
Time = 0.134 seconds
Angular velocity acceleration (α) = (0 radians/second - 565.48 radians/second) / 0.134 seconds
≈ -4223.19 radians/second^2 (rounded to two decimal places)
Therefore, the disk's angular velocity acceleration is approximately -4223.19 radians/second^2, and it takes about 0.134 seconds to stop.
To find the disk's angular velocity acceleration, we need to determine the change in angular velocity and the time it takes for this change to occur.
First, let's find the initial angular velocity of the disk. We are given that the disk spins at 5400 rpm (revolutions per minute). We need to convert this to radians per second, as angular velocity is typically measured in radians per second.
1 revolution = 2π radians
To convert from rpm to radians per second, we can use the following conversion factor:
1 rpm = 2π/60 radians per second
So, the initial angular velocity of the disk can be calculated as follows:
Initial Angular Velocity = (5400 rpm) * (2π/60 radians per second)
= 360π radians per second
Now, let's find the final angular velocity of the disk. We know that the disk completes 12 revolutions before coming to a stop. Therefore, the final angular position of the disk is 12 revolutions. We need to calculate the corresponding final angular velocity.
Final Angular Velocity = (Final Angular Position - Initial Angular Position) / Time
We can assume that the initial angular position of the disk is 0 revolutions, as it starts spinning from rest. Now, we can calculate the final angular velocity:
Final Angular Position = 12 revolutions
Final Angular Velocity = (12 revolutions - 0 revolutions) / Time
To find the time it takes to stop, we can rearrange the equation as follows:
Time = (Final Angular Position - Initial Angular Position) / Final Angular Velocity
= (12 revolutions - 0 revolutions) / (Final Angular Velocity)
To solve for the time, we need to know the final angular velocity. However, the problem does not provide this information. Without the final angular velocity, we cannot find the exact time it takes to stop the disk.
Va = 5400rev/min*1min/60s*6.28rad/rev =
565.2 rad/s.
t = (12rev/5400rev) * 60s = 0.133 s.
V = Vo + at = 0
at = -Vo
a = -Vo/t = -565/0.133 = -4248 rad/s^2.