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Ch 03: Motion in Two or Three Dimensions
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 03: Motion in Two or Three DimensionsProblem 23b
Chapter 3, Problem 23b

The earth has a radius of 6380 km and turns around once on its axis in 24 h. If arad at the equator is greater than g, objects will fly off the earth's surface and into space. (We will see the reason for this in Chapter 5.) What would the period of the earth's rotation have to be for this to occur?

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First, understand that the problem involves centripetal acceleration at the Earth's equator. The centripetal acceleration \( a_{rad} \) is given by the formula \( a_{rad} = \frac{v^2}{r} \), where \( v \) is the tangential velocity and \( r \) is the radius of the Earth.
Next, calculate the tangential velocity \( v \) at the equator using the formula \( v = \frac{2\pi r}{T} \), where \( T \) is the period of rotation. Here, \( r = 6380 \) km, which needs to be converted to meters for consistency in units.
Set the centripetal acceleration \( a_{rad} \) equal to the gravitational acceleration \( g \), which is approximately \( 9.8 \text{ m/s}^2 \). This is the condition for objects to fly off the Earth's surface.
Substitute \( v = \frac{2\pi r}{T} \) into \( a_{rad} = \frac{v^2}{r} \) and set \( a_{rad} = g \). This gives \( g = \frac{(\frac{2\pi r}{T})^2}{r} \). Simplify this equation to solve for \( T \).
Rearrange the equation to find \( T \), which is the period of rotation required for \( a_{rad} \) to be greater than \( g \). The final expression will be \( T = \frac{2\pi r}{\sqrt{gr}} \). Calculate \( T \) using the known values of \( r \) and \( g \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the circle. It is given by the formula arad = v^2/r, where v is the tangential velocity and r is the radius of the circular path. At the Earth's equator, this acceleration is due to the Earth's rotation.
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Gravitational Acceleration

Gravitational acceleration, denoted as g, is the acceleration due to Earth's gravity, approximately 9.81 m/s² at the surface. It acts downward towards the center of the Earth. For objects to remain on the surface, gravitational acceleration must be greater than the centripetal acceleration caused by Earth's rotation.
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Rotational Period

The rotational period is the time it takes for an object to complete one full rotation around its axis. For Earth, this period is 24 hours. If the rotational period decreases, the tangential velocity at the equator increases, leading to higher centripetal acceleration. The question asks for the period at which this acceleration would exceed gravitational acceleration, causing objects to fly off.
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Related Practice
Textbook Question

At its Ames Research Center, NASA uses its large '20-G' centrifuge to test the effects of very large accelerations ('hypergravity') on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge's arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5g. How fast must the astronaut's head be moving to experience this maximum acceleration?

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Textbook Question

A model of a helicopter rotor has four blades, each 3.40 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at 550 rev/min. What is the radial acceleration of the blade tip expressed as a multiple of g?

Textbook Question

A model of a helicopter rotor has four blades, each 3.40 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at 550 rev/min. What is the linear speed of the blade tip, in m/s?

Textbook Question

The earth has a radius of 6380 km and turns around once on its axis in 24 h. What is the radial acceleration of an object at the earth's equator? Give your answer in m/s2 and as a fraction of g.

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Textbook Question

A 124 kg balloon carrying a 22 kg basket is descending with a constant downward velocity of 20.0 m/s. A 1.0 kg stone is thrown from the basket with an initial velocity of 15.0 m/s perpendicular to the path of the descending balloon, as measured relative to a person at rest in the basket. That person sees the stone hit the ground 5.00 s after it was thrown. Assume that the balloon continues its downward descent with the same constant speed of 20.0 m/s. How high is the balloon when the rock is thrown?

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Textbook Question

A 124 kg balloon carrying a 22 kg basket is descending with a constant downward velocity of 20.0 m/s. A 1.0 kg stone is thrown from the basket with an initial velocity of 15.0 m/s perpendicular to the path of the descending balloon, as measured relative to a person at rest in the basket. That person sees the stone hit the ground 5.00 s after it was thrown. Assume that the balloon continues its downward descent with the same constant speed of 20.0 m/s. At the instant the rock hits the ground, how far is it from the basket?

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