Find the linear speed of the bottom of a test tube in a centrifuge if the centripetal acceleration there is 5.4×104 times the acceleration of gravity. The distance from the axis of rotation to the bottom of the test tube is 7.9cm.

V^2/R = 5.4*10^4 g = 5.29*10^5 m/s^2

R = 0.079 m

Solve for V

To find the linear speed of the bottom of the test tube in a centrifuge, we can use the formula:

Linear Speed = Radius × Angular Speed

First, let's find the angular speed using the centripetal acceleration. The formula relating centripetal acceleration (ac), angular speed (ω), and radius (r) is:

ac = ω²r

Given that the centripetal acceleration is 5.4×10⁴ times the acceleration of gravity (g), we can write:

ac = 5.4×10⁴g

Rearranging the formula, we have:

ω = √(ac / r)

Substituting the given values, where the distance from the axis of rotation (r) is 7.9 cm (or 0.079 m), and the acceleration due to gravity (g) is approximately 9.8 m/s², we get:

ω = √((5.4×10⁴ × 9.8) / 0.079)

Next, we can find the linear speed by multiplying the angular speed (ω) by the radius (r):

Linear Speed = ω × r

Substituting the values we found, we get:

Linear Speed = (√((5.4×10⁴ × 9.8) / 0.079)) × 0.079

Simplifying the equation, we have:

Linear Speed ≈ 294.8 m/s

Therefore, the linear speed of the bottom of the test tube in the centrifuge is approximately 294.8 m/s.

To find the linear speed of the bottom of the test tube in a centrifuge, you can use the equation:

v = ω * r

where v is the linear speed, ω is the angular speed, and r is the distance from the axis of rotation to the bottom of the test tube.

To find the angular speed (ω), we need to calculate the centripetal acceleration (a) using the acceleration of gravity (g) and the given ratio:

a = 5.4 × 10^4 * g

Next, we can find the angular speed using the equation:

a = ω^2 * r

Substituting the known values, we have:

5.4 × 10^4 * g = ω^2 * 0.079 m

Rearranging the equation to solve for ω:

ω^2 = (5.4 × 10^4 * g) / 0.079

Taking the square root of both sides gives:

ω = √((5.4 × 10^4 * g) / 0.079)

Now that we have the angular speed, we can find the linear speed (v):

v = ω * 0.079 m

Substituting the value of ω, we find:

v = (√((5.4 × 10^4 * g) / 0.079)) * 0.079

Using the given acceleration of gravity (g = 9.8 m/s^2), we can now calculate the linear speed.