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Chapter 7: Problem 28

The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity. (A) Will be directed towards the centre but not the same everywhere. (B) Will have the same value everywhere but not directed towards the centre. (C) Will be same everywhere in magnitude directed towards the centre. (D) Cannot be zero at any point.

Short Answer

Expert verified
The acceleration due to gravity on the surface of the Earth will be directed towards the center but not the same everywhere (A) due to uneven mass distribution in the Earth's interior.

Step by step solution

01

Understanding the Effects of Uneven Density

Since the Earth's interior is not evenly distributed, the gravitational pull caused by this uneven mass distribution will vary at different points on the surface.
02

Gravitational Force Formula

The gravitational force experienced by an object due to Earth's interior is determined by Newton's law of gravitation which states that: \[F = G \frac{m1.m2}{r^2}\] where: F : Gravitational force G : Gravitational constant, approximately \(6.674 脳 10^{-11} N路(m/kg)^2\) m1 : The mass of object 1 (in our case, a portion of the Earth's mass) m2 : The mass of object 2 (the object on the surface) r : The distance between the two objects, in our case, Earth's radius.
03

Calculate the Acceleration due to Gravity

To determine the acceleration due to gravity experienced by an object on the Earth's surface (g), we will use the gravitational force formula (F) and divide it by the mass of the object (m2): \[g = \frac{F}{m2} = G \frac{m1}{r^2}\] As the density and distribution of the Earth's mass (m1) will vary, so will the acceleration due to gravity (g), on different points on the Earth's surface.
04

Direction of Acceleration due to Gravity

The direction of the acceleration due to gravity will always be towards the center of the Earth. This is because mass attracts mass, and any object on Earth's surface will feel attraction towards the Earth's center.
05

Compare with Given Options

Let's compare our findings with the given options: (A) Will be directed towards the centre but not the same everywhere. --> Correct. As we have shown, due to the uneven mass distribution, the value of acceleration due to gravity will vary across different points on Earth's surface, but its direction will always be towards the center. (B) Will have the same value everywhere but not directed towards the centre. ---> Incorrect. The direction will be towards the center, but the value will not be the same everywhere. (C) Will be same everywhere in magnitude directed towards the centre. ---> Incorrect. While the direction is correct, the value will not be the same everywhere. (D) Cannot be zero at any point. ---> Correct. It is highly improbable for the acceleration due to gravity to be zero at any point on Earth's surface, since mass attracts mass and there will always be some gravitational pull experienced. Conclusion: The correct answer is (A), the acceleration due to gravity will be directed towards the center but not the same everywhere on the surface of the Earth.

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Most popular questions from this chapter

From a solid sphere of mass \(M\) and radius \(R\), a spherical portion of radius \(\frac{R}{2}\) is removed, as shown in Fig. 7.17. Taking gravitational potential \(V=0\) at \(r=\infty\), the potential at the centre of the cavity thus formed is \((G=\) gravitational constant) (A) \(\frac{-G M}{R}\) (B) \(\frac{-2 G M}{3 R}\) (C) \(\frac{-2 G M}{R}\) (D) \(\frac{-G M}{2 R}\)

A person brings a mass of \(1 \mathrm{~kg}\) from infinity to a point \(A\). Initially the mass was at rest but it moves at a speed of \(2 \mathrm{~m} / \mathrm{s}\) as it reaches \(A\). The work done by the person on the mass is \(-3 \mathrm{~J}\). The potential at \(A\) is (A) \(-3 \mathrm{~J} / \mathrm{kg}\) (B) \(-2 \mathrm{~J} / \mathrm{kg}\) (C) \(-5 \mathrm{~J} / \mathrm{kg}\) (D) None of these

A uniform spherical shell gradually shrinks maintaining its shape. The gravitational potential at the centre (A) Increases (B) Decreases (C) Remains constant (D) Oscillates

Consider an attractive force which is central but is inversely proportional to the first power of distance. If such a particle is in circular orbit under such a force, which of the following statements are correct. (A) \(v \propto r\) (B) \(v \propto r^{\circ}\) (C) \(T \propto r\) (D) \(T \propto r^{\circ}\)

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