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Chapter 6: Problem 719

In a gravitational field, at a point where the gravitational potential is zero (A) The gravitational field is necessarily zero (B) The gravitational field is not necessarily zero (C) Nothing can be said definitely, about the gravitational field (D) None of these

Short Answer

Expert verified
In a gravitational field, at a point where the gravitational potential is zero, the gravitational field is the negative derivative of the gravitational potential with respect to distance. As no information about the behavior of gravitational potential around that point is given, nothing can be said definitely about the gravitational field in this situation. Hence, the correct answer is (C) Nothing can be said definitely about the gravitational field.

Step by step solution

01

Gravitational Potential and Gravitational Field Formula

In a gravitational field, the gravitational potential (V) is the amount of work done per unit mass to bring an object from infinity to a certain point. The gravitational field (g) represents the force experienced by a unit mass placed at that point. Mathematically, these are represented as: \(V = -\dfrac{GM}{r}\) \(g = \dfrac{GM}{r^2}\) Where \(G\) is the gravitational constant, \(M\) is the mass of the object producing the gravitational field, and \(r\) is the distance from the center of the object.
02

Relationship Between Gravitational Potential and Gravitational Field

Now, we know that the gravitational potential is zero at the given point. We need to find out the behavior of the gravitational field at this point. To do so, let's differentiate the gravitational potential with respect to distance (\(r\)): \(\dfrac{dV}{dr} = \dfrac{d\left(-\dfrac{GM}{r}\right)}{dr} = \dfrac{GM}{r^2} \) \(g = -\dfrac{dV}{dr}\) This equation shows that the gravitational field is the negative derivative of the gravitational potential with respect to distance.
03

Evaluating the Gravitational Field in the Given Situation

According to the given exercise, the gravitational potential is zero at a certain point. It doesn't provide any information about the behavior of gravitational potential around that point. So, we cannot determine if the gravitational field is zero or nonzero at that point. To answer this question, we need more information about the behavior of gravitational potential around that point. Therefore, the correct answer is: (C) Nothing can be said definitely about the gravitational field.

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Most popular questions from this chapter

The time period \(\mathrm{T}\) of the moon of planet Mars \((\mathrm{Mm})\) is related to its orbital radius \(\mathrm{R}\) as \((\mathrm{G}=\) Gravitational constant \()\) (A) $\mathrm{T}^{2}=\left[\left(4 \pi^{2} \mathrm{R}^{3}\right) /(\mathrm{GMm})\right]$ (B) $\mathrm{T}^{2}=\left[\left(4 \pi^{2} \mathrm{GR}^{3}\right) /(\mathrm{Mm})\right]$ (C) \(T^{2}=\left[\left(2 \pi R^{2} G\right) /(M m)\right]\) (D) \(\mathrm{T}^{2}=4 \pi \mathrm{Mm} \mathrm{GR}^{2}\)

On the surface of earth acceleration due to gravity is \(\mathrm{g}\) and gravitational potential is \(\mathrm{V}\) match the followingTable - 1 Table \(-2\) (A) At height \(\mathrm{h}=\mathrm{R}\) value of \(\mathrm{g}\) (P) decrease by a factor \((1 / 4)\) (B) At depth \(\mathrm{h}=(\mathrm{R} / 2)\) (Q) decrease by a factor \((1 / 2)\) (C) At height \(\mathrm{h}=\mathrm{R}\) value of \(\mathrm{v}\) (R) increase by a factor \((11 / 8)\) (D) At depth \(\mathrm{h}=(\mathrm{R} / 2)\) value of \(\mathrm{v}\) (S) increase by a factor 2 (T) None

Two bodies of masses \(m_{1}\) and \(m_{2}\) are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual a gravitational attraction Their relative velocity of approach at separation distance \(\mathrm{r}\) between them is (A) $\left[\left\\{2 \mathrm{G}\left(\mathrm{m}_{1}-\mathrm{m}_{2}\right)\right\\} / \mathrm{r}\right]^{-1 / 2}$ (B) $\left[\left\\{2 \mathrm{G}\left(\mathrm{m}_{1}+\mathrm{m}_{2}\right)\right\\} / \mathrm{r}\right]^{1 / 2}$ (C) $\left[\mathrm{r} /\left\\{2 \mathrm{G}\left(\mathrm{m}_{1} \mathrm{~m}_{2}\right)\right\\} / \mathrm{r}\right]^{1 / 2}$ (D) $\left[\left(2 \mathrm{Gm}_{1} \mathrm{~m}_{2}\right) / \mathrm{r}\right]^{1 / 2}$

Gravitational potential at any point inside a spherical shall is uniform and is given by \(-(\mathrm{GM} / \mathrm{R})\) where \(\mathrm{M}\) is the mass of shell and \(\mathrm{R}\) its radius. At the center solid sphere, potential is \([-\\{(3 \mathrm{GM}) /(2 \mathrm{R})\\}]\)(1) There is a concentric hole of radius \(\mathrm{R}\) in a solid sphere of radius \(2 \mathrm{R}\) mass of the remaining portion is \(\mathrm{M}\) what is the gravitational at center? (A) \(-[(3 \mathrm{GM}) / 7 \mathrm{R}]\) (B) \(-[(5 \mathrm{GM}) / 7 \mathrm{R}]\) (C) \(-[(7 \mathrm{GM}) / 14]\) (D) \(-[(9 \mathrm{GM}) /(14 \mathrm{R})]\)

orbital velocity of earth's satellite near the surface is $7 \mathrm{kms}^{-1}$. when the radius of orbit is 4 times that of earth's radius, then orbital velocity in that orbit is $=\ldots \ldots \ldots \ldots \mathrm{kms}^{-1}$ (A) \(3.5\) (B) 17 (C) 14 (D) 35
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