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Chapter 8: Problem 1

Which of the following statements is/are true? For the particle on the surface of earth (a) the linear speed is maximum at the poles (b) the linear speed is minimum at the equator (c) the angular speed is maximum at the equator (d) the angular speed is \(7.3 \times 10^{-5} \mathrm{rad} / \mathrm{s}\) at the equator

Short Answer

Expert verified
Only statement (d) is true.

Step by step solution

01

Understand the Problem

To determine which of the given statements are true, we need to discuss linear and angular speed of a point on the Earth's surface as it rotates. Linear speed depends on distance from the axis of rotation, while angular speed is constant for any point on the Earth.
02

Analyze Statement (a)

Statement (a) claims that linear speed is maximum at the poles. Since linear speed depends on the distance from the axis, and a point at the pole is right on the axis (distance = 0), linear speed is actually zero at the poles.
03

Analyze Statement (b)

Statement (b) asserts that linear speed is minimum at the equator. The linear speed is determined by multiplying angular speed with the radius from the axis to the surface point. At the equator, this distance is maximum, hence linear speed there is also maximum, making this statement false.
04

Analyze Statement (c)

Statement (c) states that angular speed is maximum at the equator. However, angular speed is the same everywhere on Earth because it depends only on Earth's rotation rate. Thus, this statement is false.
05

Analyze Statement (d)

Statement (d) claims angular speed is \( 7.3 \times 10^{-5} \ \mathrm{rad} / \mathrm{s} \) at the equator. The angular speed of Earth, derived from its rotational period (one revolution per day), is indeed approximately \( 7.3 \times 10^{-5} \ \mathrm{rad/s} \). This statement is true.

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Key Concepts

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

Linear Speed
Linear speed refers to how fast a point on the Earth's surface is moving as the Earth rotates. This speed is dependent on the distance from the Earth's axis of rotation. Imagine a merry-go-round. Points further from the center move faster than those near the middle. The same applies to Earth.
For any point on Earth:
  • Linear speed is zero at the poles because these points are exactly on the axis of rotation.
  • Linear speed is highest at the equator, as it's farthest from the axis.
Knowing this, we can debunk the myth that the poles have the fastest linear speed. It's actually the equator.
Rotation of Earth
The Earth's rotation is the spinning of the planet on its axis. It completes one full rotation approximately every 24 hours, leading to day and night cycles.
This rotation is responsible for several noticeable effects:
  • Regular alteration of day and night.
  • Constant linear speed variations based on latitude.
  • Consistent angular speed globally.
Despite the linear speed changes across different latitudes, the angular speed, unlike linear speed, remains constant wherever you are on Earth due to the uniform rotational period.
Poles
The poles are unique points on Earth where the axis of rotation intersects the surface. There are two poles: the North Pole and the South Pole. Because they are on the axis itself:
  • Their linear speed is zero since they don't cover any distance in rotation.
  • You'd be standing in place as the world turns, not spinning around like you would at the equator.
This makes poles interesting yet calm places if you're interested in covering the least possible distance while Earth rotates.
Equator
The equator is an imaginary line circling the Earth, equidistant from both poles. It stands out for several reasons:
  • It is the widest part of the Earth, thus farthest from the axis.
  • This makes linear speed at the equator the highest due to the maximum radius.
Moreover, while the linear speed peaks here, the angular speed remains the same as elsewhere on the planet. This combination of factors explains why high-speed phenomena are especially intense at the equator, in contrast to the stillness of the poles.

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Most popular questions from this chapter

If earth rotates \(n\) time faster than its present speed \(\omega\) about its axis in order that the bodies lying on the equator of earth just fly off into the space, then the value of \(n\) is equal to (take radius of earth \(R\) ) (a) \(\omega \sqrt{g / R}\) (b) \(\frac{1}{\omega} \sqrt{\frac{g}{R}}\) (c) \(\omega \sqrt{2 g / R}\) (d) \(\omega \sqrt{g / 2 R}\)

Three uniform spheres each having of mass \(M\) and radius \(a\) are kept in such a way that they touch each other. The magnitude of the gravitational force on any one of the spheres, due to the other two is (a) zero (b) \(\frac{\sqrt{3} G M^{2}}{4 a^{2}}\) (c) \(\frac{3 G M^{2}}{2 a^{2}}\) (d) \(\frac{\sqrt{2} G M^{2}}{a^{2}}\)

Astronauts inside the satellite are always in the state of weightlessness. The reason behind this, is (a) there is no gravitational force acting on them (b) the gravitational force of earth balances that of the sun (c) there is no atmosphere at the height at which they are orbiting (d) their weight counter balance with the force directed away from the centre of the planet round which the satellite is orbiting

The earth-moon distance is \(3.8 \times 10^{5} \mathrm{~km}\) and mass of earth is 81 times that of moon. The distance from the earth where the gravitation field due to earth and moon cancel out is (a) \(1.42 \times 10^{5} \mathrm{~km}\) (b) \(2.42 \times 10^{5} \mathrm{~km}\) (c) \(3.42 \times 10^{5} \mathrm{~km}\) (d) \(10^{5} \mathrm{~km}\)

At a given place where acceleration due to gravity is \(g \mathrm{~m} / \mathrm{s}^{2}\) a sphere of lead of density \(\rho \mathrm{kg} / \mathrm{m}^{3}\) is gently released in a column of non-viscous liquid of density \(\sigma \mathrm{kg} / \mathrm{m}^{3}\). If \(\rho>\sigma\), then the acceleration of the sphere is (a) \(g\) (b) zero (c) \(\left(\frac{\rho-\sigma}{\rho}\right) g\) (d) \(\frac{\sigma}{\rho} g\)
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