Find the linear speed of the bottom of a test tube in a centrifuge if the centripetal acceleration there is 5.1×104 times the acceleration of gravity. The distance from the axis of rotation to the bottom of the test tube is 7.7 .
To find the linear speed of the bottom of the test tube in a centrifuge, we can use the formula:
Linear speed (v) = Radius (r) x Angular speed (ω)
First, let's find the angular speed (ω) using the given information. We know that the centripetal acceleration (a) is 5.1 x 10^4 times the acceleration due to gravity (g). The formula for centripetal acceleration is:
a = rω^2
Where r is the distance from the axis of rotation to the bottom of the test tube.
We can rearrange this formula to solve for ω:
ω = sqrt(a/r)
Substituting the given values:
ω = sqrt((5.1 x 10^4) * g / r)
Next, we need to convert the distance from the axis of rotation (r) from meters to centimeters since acceleration due to gravity (g) is given in cm/s^2.
r = 7.7 cm
Now, we need to convert the acceleration due to gravity (g) from m/s^2 to cm/s^2:
g = 9.8 m/s^2 * 100 cm/m = 980 cm/s^2
Substituting the values into the formula:
ω = sqrt((5.1 x 10^4) * 980 cm/s^2 / 7.7 cm)
Calculating this expression gives us the value of ω.
Once we have the value of ω, we can plug it back into the formula for linear speed (v) to find the final answer.