If the rotational speed of a CD changes from 650 rpm to 250 rpm as it plays 30 minutes of music, what is the CD's average rotational acceleration?
To find the CD's average rotational acceleration, we need to calculate the change in rotational speed and the time taken for this change.
First, let's calculate the change in rotational speed:
Change in rotational speed = Final rotational speed - Initial rotational speed
= 250 rpm - 650 rpm
= -400 rpm
Next, let's convert the change in rotational speed to radians per second (since the unit of rotational speed should be in radians per second for this calculation):
Change in rotational speed in radians per second = Change in rotational speed in rpm * (2π radians / 1 minute / 60 seconds)
= -400 rpm * (2π radians / 1 minute / 60 seconds)
= -400 rpm * (2π radians / 1 minute / 60 seconds)
= -40π radians per second
Now, we need to find the time taken for this change. We are given that the CD plays music for 30 minutes. Since the unit of time should be in seconds for this calculation, we multiply by 60:
Time taken = 30 minutes * 60 seconds / 1 minute
= 1800 seconds
Finally, we can calculate the average rotational acceleration using the formula:
Average rotational acceleration = Change in rotational speed in radians per second / Time taken
= (-40π radians per second) / (1800 seconds)
≈ -0.0707 radians per second squared
So, the CD's average rotational acceleration is approximately -0.0707 radians per second squared.
To determine the CD's average rotational acceleration, we need to use the formula:
Average rotational acceleration (α) = (ωf - ωi) / t
Where:
- α is the average rotational acceleration
- ωf is the final rotational speed (in radians per minute)
- ωi is the initial rotational speed (in radians per minute)
- t is the time interval (in minutes)
In this case, the final rotational speed ωf is 250 rpm, the initial rotational speed ωi is 650 rpm, and the time interval t is 30 minutes.
First, let's convert the rotational speeds from rpm to radians per minute. Since 1 revolution is equal to 2π radians, we can use the conversion factor:
1 revolution = 2π radians
So, for example, 650 rpm can be converted to radians per minute as follows:
650 rpm = 650 revolutions per minute
= 650 * 2π radians per minute
Now, let's calculate the average rotational acceleration using the formula:
Average rotational acceleration (α) = (ωf - ωi) / t
α = (250 * 2π - 650 * 2π) / 30
Simplifying this expression gives us:
α = (-400π) / 30
To find the numerical value of α, we need to calculate this expression:
α ≈ -41.89 radians per minute^2
Therefore, the CD's average rotational acceleration is approximately -41.89 radians per minute^2.
changespeed = 2PI(350-650)/60 rad/sec
change in time= 30*60 sec
acceleration= changespeed/time= 2PI(350-650)/(30*60*60)