Each of the space shuttle's main engines is fed liquid hydrogen by a high-pressure pump. Turbine blades inside the pump rotate at 595 rev/s. A point on one of the blades traces out a circle with a radius of 0.02 m as the blade rotates.
(a) What is the magnitude of the centripetal acceleration that the blade must sustain at this point?
1 m/s2
(b) Express this acceleration as a multiple of g = 9.80m/s2.
(a) The centripetal acceleration can be calculated using the formula:
a = rω²
where r is the radius and ω is the angular velocity.
Here, the radius (r) is 0.02 m and the angular velocity (ω) is given as 595 rev/s.
First, we need to convert the angular velocity from rev/s to rad/s:
ω = 595 rev/s * (2π rad/rev) ≈ 3738 rad/s
Plugging in the values:
a = (0.02 m) * (3738 rad/s)²
a ≈ 56.2 m/s²
The magnitude of the centripetal acceleration is approximately 56.2 m/s².
(b) To express this acceleration as a multiple of g = 9.80 m/s², we can divide the centripetal acceleration by the acceleration due to gravity:
a/g = (56.2 m/s²) / (9.80 m/s²)
a/g ≈ 5.7
So, the magnitude of the centripetal acceleration is approximately 5.7 times the acceleration due to gravity.
To determine the centripetal acceleration, we can use the formula for centripetal acceleration:
a = rω²
where:
a = centripetal acceleration
r = radius of the circle
ω = angular velocity
(a) Given that the radius of the circle is 0.02 m and the angular velocity is 595 rev/s, we can find the centripetal acceleration by substituting these values into the formula:
a = (0.02 m)(595 rev/s)²
First, let's convert the angular velocity from rev/s to rad/s. One revolution is equal to 2π radians, so:
ω = 595 rev/s × (2π rad/1 rev) = 595 × 2π rad/s
Substituting this value into the formula, we have:
a = (0.02 m)(595 × 2π rad/s)²
Simplifying further, we get:
a = (0.02 m)(595 × 2π rad/s)² = (0.02 m)(595 × 2π rad/s)(595 × 2π rad/s)
Using a calculator or computational tool to calculate this expression, we find:
a ≈ 1661.516 m/s²
Therefore, the magnitude of the centripetal acceleration that the blade must sustain at this point is approximately 1661.516 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:
a/g = (1661.516 m/s²) / (9.8 m/s²)
Calculating this ratio, we find:
a/g ≈ 169.864
Therefore, the acceleration is approximately 169.864 times the acceleration due to gravity, g.
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
r is the radius of the circle
(a) First, we need to convert the angular velocity from revolutions per second to radians per second. To do that, we use the conversion factor:
1 revolution = 2π radians
So, the angular velocity in radians per second is:
ω = 595 rev/s * 2π rad/rev
ω = 1190π rad/s
Now we can calculate the centripetal acceleration:
a = (1190π rad/s)^2 * 0.02 m
a ≈ 141208π m/s^2
(b) To express this acceleration as a multiple of g, we divide it by the acceleration due to gravity, g.
a/g = (141208π m/s^2) / 9.8 m/s^2
a/g ≈ 14396π
Therefore, the centripetal acceleration is approximately 14396π times greater than the acceleration due to gravity (g).
Centripetal acceleration is given by
a = v^2/r = omega^2*r
where a is the centripetal acceleration, v is the speed, r is the radius, and omega is the angular speed in rad/s
Convert rev/s to radians per second
595 rev/s * (2*pi radians/rev) = ____ = omega
a = omega^2 * 0.02 m
b). Divide the centripetal acceleration by g to express it as a multiple of g.