A laboratory centrifuge on earth makes n rpm and produces an acceleration of 3.20 g at its outer end.
A) What is the acceleration (in g's, i.e., acceleration divided by g) at a point halfway out to the end?
B) This centrifuge is now used in a space capsule on the planet Mercury, where g_mercury is 0.378 what it is on earth. How many rpm (in terms of n) should it make to produce 7 g_mercury at its outer end?
Ac = v^2/R = w^2 r
divide r by two --> divide Ac by to
7*.378 = 2.646 g
so
Ac final = 2.646/3.2 = .827 Ac original
w^2 r final = .827 w^2 r original
r did not change
w final = sqrt (.827) w original
n final = .91 n original
can you explain how to find the acceleration at a point halfway out to the end?
sure,
Ac = w^2, the angular velocity squared
times the radius.
If you cut the radius in half, go halfway out, then you cut the acceleration in half since w is the same.
So how do you get the first part? "{
3.2 / 2
A) To find the acceleration at a point halfway out to the end in terms of g's, we can use the concept of centripetal acceleration. The acceleration due to gravity, which is typically denoted as g, can be used as a gravitational constant in this scenario.
Given that the centrifuge produces an acceleration of 3.20 g at its outer end, we can calculate the acceleration at the halfway point as follows:
- The acceleration at the outer end is 3.20 g.
- The acceleration at the halfway point will be half of the acceleration at the outer end.
- So the acceleration at the halfway point is (3.20 g) / 2.
B) To find the number of rpm in terms of n needed to produce 7 g_mercury at the outer end on the planet Mercury, we need to take into account the different value of g on Mercury.
Given that g_mercury is 0.378 times what it is on Earth, we can set up the equation as follows:
- The acceleration at the outer end on Mercury is (7 g_mercury).
- The acceleration due to gravity on Mercury is (g_mercury * g).
- So, (7 g_mercury) = (g_mercury * g).
To isolate g_mercury, we divide both sides of the equation by g:
- (7 g_mercury) / g = g_mercury.
Therefore, to produce 7 g_mercury at the outer end on Mercury, the centrifuge should make (7 g_mercury) rpm. Since (g_mercury) = 0.378 * (g), we can multiply both sides of the equation by 0.378:
- (7 g_mercury) rpm = 0.378 * 7 * n rpm.
Simplifying further:
- (7 g_mercury) rpm = 2.646 n rpm.
So, the centrifuge should make 2.646 n rpm to produce 7 g_mercury at its outer end on the planet Mercury.