Each of the space shuttle's main engines is fed liquid hydrogen by a high-pressure pump. Turbine blades inside the pump rotate at 635 rev/s. A point on one of the blades traces out a circle with a radius of 0.021 m as the blade rotates.
(a) What is the magnitude of the centripetal acceleration that the blade must sustain at this point?
(b) Express this acceleration as a multiple of g = 9.80m/s2.
To find the magnitude of the centripetal acceleration, we can use the formula:
a = ω^2 * r
where:
a is the centripetal acceleration,
ω (omega) is the angular velocity in radians per second,
and r is the radius of the circular path.
Given:
ω = 635 rev/s = 635 * 2π rad/s (since 1 revolution = 2π radians)
r = 0.021 m
(a) Substituting the given values, we have:
a = (635 * 2π rad/s)^2 * 0.021 m
To calculate this, we first calculate the square of the angular velocity:
(635 * 2π rad/s)^2 = (4,000π)^2 rad^2/s^2 = 16,000,000π^2 rad^2/s^2
Now, multiplying by the radius:
a = 16,000,000π^2 rad^2/s^2 * 0.021 m ≈ 338,904.410069 m/s^2
Therefore, the magnitude of the centripetal acceleration that the blade must sustain at this point is approximately 338,904.410069 m/s^2.
(b) To express this acceleration as a multiple of g = 9.80 m/s^2, we can divide the centripetal acceleration by the acceleration due to gravity.
a/g = 338,904.410069 m/s^2 / 9.80 m/s^2 ≈ 34,581.951 m/s^2
So, the centripetal acceleration at this point is approximately 34,581.951 times the acceleration due to gravity.