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Chapter 5: Problem 57

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 \(0.020 \mathrm{m}\) 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 \(g=9.80 \mathrm{m} / \mathrm{s}^{2}\).

Short Answer

Expert verified
(a) The centripetal acceleration is \(4.76 \times 10^5 \text{ m/s}^2\). (b) This is approximately \(4.86 \times 10^4\) times \( g \).

Step by step solution

01

Find Angular Velocity

To find the angular velocity \( \omega \) in radians per second, use the conversion from revolutions per second. Given that the turbine blades rotate at 617 rev/s, use the formula \( \omega = 2\pi f \), where \( f \) is the frequency in rev/s. Thus, \( \omega = 2\pi \times 617 \).
02

Calculate Centripetal Acceleration

The centripetal acceleration \( a_c \) can be calculated using the formula \( a_c = \omega^2 r \), where \( r \) is the radius of the circle traced by the point on the blade. We have \( r = 0.020 \text{ m} \) and \( \omega = 2\pi \times 617 \). Substitute these values to find \( a_c \).
03

Express Centripetal Acceleration as a Multiple of g

To express the centripetal acceleration \( a_c \) as a multiple of \( g \), divide the centripetal acceleration by \( g = 9.80 \text{ m/s}^2 \). Compute the multiple \( \frac{a_c}{g} \) to find how many times larger \( a_c \) is compared to the acceleration due to gravity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Velocity
Angular velocity is the rate at which an object rotates around a particular axis. It's often used when discussing objects moving in circular paths, such as turbine blades. Angular velocity is measured in radians per second (rad/s). It helps us understand how quickly a point on a rotating object moves around a circle.

The formula to find angular velocity is crucial:
  • Convert frequency in revolutions per second (rev/s) to angular velocity using the formula \( \omega = 2\pi f \), where \( f \) is the frequency.
In this exercise, the turbine blade rotates at 617 rev/s. Applying the formula, we calculate: - \( \omega = 2\pi \times 617 \). This value gives us the speed at which the point travels around the circle formed by the blade's rotation.
Turbine Blades
Turbine blades are essential components in high-pressure pumps, like those used in space shuttle engines. They rotate rapidly to propel fluids like liquid hydrogen effectively. The speed and precision of these blades are critical for the engine's functionality.

To understand them further, here are some key points:
  • They are engineered to handle extreme forces while maintaining efficiency.
  • They often rotate at thousands of revolutions per minute.
  • Each blade traces a circular path, crucial for the calculations of centripetal forces.
In the context of this exercise, a point on a turbine blade traces a circle of radius \(0.020 \, m\). This path is vital to determine the forces in play, which include centripetal acceleration.
Space Shuttle Engines
Space shuttle engines are marvels of engineering, requiring many complex components working harmoniously. Among these components, the liquid hydrogen fuel system powered by high-speed turbine blades plays a critical role.

Key features of space shuttle engines include:
  • They use high-pressure pumps to deliver fuel efficiently.
  • These engines must withstand extreme temperatures and pressures.
  • Every component, such as the turbine blades, needs precise calibration to ensure optimal performance.
Understanding how each part operates, including the rotational dynamics of the turbine blades, provides insight into the engine's performance capabilities.
Physics Calculations
Physics calculations are essential for solving problems involving motion, particularly in scenarios with rotating objects. These calculations help us determine velocities and accelerations in circular paths.

Some important aspects of physics calculations in this context include:
  • Calculating centripetal acceleration, using the formula \( a_c = \omega^2 r \), where \( \omega \) is angular velocity and \( r \) is the radius.
  • Expressing acceleration as a multiple of the acceleration due to gravity, \( g = 9.80 \, m/s^2 \).
By substituting \( \omega = 2\pi \times 617 \) and \( r = 0.020 \, m \) into the centripetal acceleration formula, we find the value of \( a_c \), and further divide by \( g \) to express it as a multiple of gravity. These calculations reinforce our understanding of motion and forces in rotational contexts.

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Most popular questions from this chapter

A stone is tied to a string (length \(=1.10 \mathrm{m}\) ) and whirled in a circle at the same constant speed in two different ways. First, the circle is horizontal and the string is nearly parallel to the ground. Next, the circle is vertical. In the vertical case the maximum tension in the string is \(15.0 \%\) larger than the tension that exists when the circle is horizontal. Determine the speed of the stone.

Computer-controlled display screens provide drivers in the Indianapolis 500 with a variety of information about how their cars are performing. For instance, as a car is going through a turn, a speed of \(221 \mathrm{mi} / \mathrm{h}\) \((98.8 \mathrm{m} / \mathrm{s})\) and centripetal acceleration of \(3.00 \mathrm{g}\) (three times the acceleration due to gravity) are displayed. Determine the radius of the turn (in meters).

A special electronic sensor is embedded in the seat of a car that takes riders around a circular loop-the-loop ride at an amusement park. The sensor measures the magnitude of the normal force that the seat exerts on a rider. The loop- the-loop ride is in the vertical plane and its raerts on a rider. The loop- the-loop ride is in the vertical plane and its radius is \(21 \mathrm{m} .\) Sitting on the seat before the ride starts, a rider is level and stationary, and the electronic sensor reads 770 N. At the top of the loop, the rider is upside down and moving, and the sensor reads \(350 \mathrm{N}\). What is the speed of the rider at the top of the loop?

A car travels at a constant speed around a circular track whose radius is 2.6 km. The car goes once around the track in 360 s. What is the magnitude of the centripetal acceleration of the car?

The aorta is a major artery, rising upward from the left ventricle of the heart and curving down to carry blood to the abdomen and lower half of the body. The curved artery can be approximated as a semicircular arch whose diameter is \(5.0 \mathrm{cm} .\) If blood flows through the aortic arch at a speed of \(0.32 \mathrm{m} / \mathrm{s},\) what is the magnitude \(\left(\mathrm{in} \mathrm{m} / \mathrm{s}^{2}\right)\) of the blood's centripetal acceleration?
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