Each of the space shuttle’s main engines is fed liquid hydrogen by a high-pressure pump. Turbine blades inside the pump rotate at 617 rev/s. A point on one of the blades traces out a circle with a radius of 0.020 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 = 9.80 m/s^2
Please post the steps you did to solve it too, thanks!
To solve this problem, we need to find the centripetal acceleration and then express it as a multiple of g.
(a) To find the magnitude of the centripetal acceleration, we can use the formula a = rω², where a is the centripetal acceleration, r is the radius, and ω is the angular velocity.
Given:
Radius (r) = 0.020 m
Angular velocity (ω) = 617 rev/s
First, we need to convert the angular velocity from rev/s to rad/s. Since 1 revolution is equal to 2π radians, we can multiply the angular velocity by 2π.
Angular velocity (ω) = 617 rev/s * 2π rad/rev = 1229.94 rad/s
Now we can calculate the centripetal acceleration (a):
a = rω² = (0.020 m) * (1229.94 rad/s)²
Calculating this gives:
a = 0.020 m * (1229.94 rad/s)² ≈ 31,064.58 m/s²
So the magnitude of the centripetal acceleration is approximately 31,064.58 m/s².
(b) To express this acceleration as a multiple of g, we divide the magnitude of the centripetal acceleration by the acceleration due to gravity (g = 9.80 m/s²).
Expressing centripetal acceleration as a multiple of g:
(31,064.58 m/s²) / (9.80 m/s²) ≈ 3,173.06
Therefore, the magnitude of the centripetal acceleration is approximately 3,173.06 times the acceleration due to gravity (g).
To find the magnitude of the centripetal acceleration that the blade must sustain at the given point, we can use the following formula:
Centripetal acceleration (a) = (angular velocity)^2 × radius
Given:
- Angular velocity (ω) = 617 rev/s (Note: We need to convert this to radians per second)
- Radius (r) = 0.020 m
To convert angular velocity from revolutions per second to radians per second, we need to multiply by 2π (since 1 revolution = 2π radians).
Angular velocity (ω) = 617 rev/s × 2π rad/rev
= 1229.94π rad/s
Substituting the values into the formula, we have:
a = (1229.94π rad/s)^2 × 0.020 m
Now, let's calculate the answer:
a ≈ (1519168.4π^2) m/s^2
≈ 1519168.4 × (3.14159)^2 m/s^2
≈ 1519168.4 × 9.87 m/s^2
≈ 14999946 m/s^2
Therefore, the magnitude of the centripetal acceleration that the blade must sustain at this point is approximately 14999946 m/s^2.
To express this acceleration as a multiple of g = 9.80 m/s^2, divide the centripetal acceleration (14999946 m/s^2) by g:
Multiple of g = 14999946 m/s^2 / 9.80 m/s^2
≈ 1530633.3
So, the acceleration is approximately 1530633.3 times the acceleration due to gravity (g).
In summary:
(a) The magnitude of the centripetal acceleration is approximately 14999946 m/s^2.
(b) The acceleration is about 1530633.3 times the acceleration due to gravity.
centripetal acceleration= w^2 r
change rev/sec to rad/sec