Suppose the centripetal acceleration of the sample is 7.95 x 103 times as large as the acceleration due to gravity. How many revolutions per minute is the sample making, if it is located at a radius of 3.08 cm from the axis of rotation?
To find the number of revolutions per minute the sample is making, we need to use the formula for centripetal acceleration:
centripetal acceleration = (angular velocity)^2 * radius
The centripetal acceleration is given as 7.95 x 10^3 times the acceleration due to gravity. We can rewrite this as:
centripetal acceleration = (7.95 x 10^3) * (acceleration due to gravity)
Since the acceleration due to gravity is approximately 9.8 m/s^2, we can substitute this value into the equation:
centripetal acceleration = (7.95 x 10^3) * (9.8 m/s^2)
Now, let's convert the radius to meters:
radius = 3.08 cm = 0.0308 m
Substituting all the values into the centripetal acceleration formula, we have:
centripetal acceleration = (7.95 x 10^3) * (9.8 m/s^2) = 7.791 x 10^4 m/s^2
Next, we need to solve for the angular velocity, which is given by:
angular velocity = sqrt(centripetal acceleration / radius)
Plugging in the values, we get:
angular velocity = sqrt((7.791 x 10^4 m/s^2) / (0.0308 m))
Calculating this gives us:
angular velocity = 9113.9 rad/s
Finally, to convert the angular velocity to revolutions per minute (rpm), we use the conversion factor:
1 revolution = 2π radians
1 minute = 60 seconds
Therefore,
1 rpm = (2π / 60) rad/s
To find the number of revolutions per minute, we divide the angular velocity by the conversion factor:
number of revolutions per minute = (9113.9 rad/s) / (2π / 60 rad/s)
Calculating this gives us:
number of revolutions per minute = 8703.4 rpm
Therefore, the sample is making approximately 8703.4 revolutions per minute.