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Chapter 5: Problem 579

If the earth is treated as a sphere of radius \(R\) and mass \(M\). Its angular momentum about the axis of rotation with period \(\mathrm{T}\) is..... \(\\{\mathrm{A}\\}\left(\pi \mathrm{MR}^{3} / \mathrm{T}\right)\) \(\\{\mathrm{B}\\}\left(\operatorname{MR}^{2} \pi / \mathrm{T}\right)\) \(\\{C\\}\left(2 \pi \mathrm{MR}^{2} / 5 \mathrm{~T}\right)\) \(\\{\mathrm{D}\\}\left(4 \pi \mathrm{MR}^{2} / 5 \mathrm{~T}\right)\)

Short Answer

Expert verified
The short answer is: The angular momentum of the Earth with given conditions is \(L = \dfrac{4\pi MR^2}{5T}\). Thus, the correct option is D.

Step by step solution

01

Recall the formula for angular momentum of a solid sphere

The formula for angular momentum (L) of a solid sphere of mass M and radius R, rotating about an axis through its center with an angular velocity ω is: \(L = \dfrac{2}{5}MR^2 ω\)
02

Find the relationship between period and angular velocity

The period (T) of rotation of a rotating object is the time taken for one complete revolution. The angular velocity (ω) is the rate of change of angular displacement. The relationship between angular velocity (ω) and period (T) can be written as: \(ω = \dfrac{2\pi}{T}\)
03

Substitute the relationship between ω and T in the angular momentum formula

Now, let's substitute the relationship between ω and T in the formula for the angular momentum of a solid sphere: \(L = \dfrac{2}{5}MR^2 \cdot \dfrac{2\pi}{T}\)
04

Simplify the expression

Simplify the expression for the angular momentum: \(L = \dfrac{4\pi MR^2}{5T}\)
05

Compare the simplified expression with the given options

Now, we compare our simplified expression with the given options: \(L = \dfrac{4\pi MR^2}{5T}\) matches with option D. So the correct answer is: \(\\{\mathrm{D}\\}\left(4 \pi \mathrm{MR}^{2} / 5 \mathrm{~T}\right)\).

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Most popular questions from this chapter

The moment of inertia of a thin rod of mass \(\mathrm{M}\) and length \(\mathrm{L}\) about an axis passing through the point at a distance $\mathrm{L} / 4$ from one of its ends and perpendicular to the rod is \(\\{\mathrm{A}\\}\left[\left(7 \mathrm{ML}^{2}\right) / 48\right]\) \\{B \\} [ \(\left[\mathrm{ML}^{2} / 12\right]\) \(\\{\mathrm{C}\\}\left[\left(\mathrm{ML}^{2} / 9\right]\right.\) \(\\{\mathrm{D}\\}\left[\left(\mathrm{ML}^{2} / 3\right]\right.\)

Statement \(-1-\) A thin uniform rod \(A B\) of mass \(M\) and length \(\mathrm{L}\) is hinged at one end \(\mathrm{A}\) to the horizontal floor initially it stands vertically. It is allowed to fall freely on the floor in the vertical plane, The angular velocity of the rod when its ends \(B\) strikes the floor $\sqrt{(3 g / L)}\( Statement \)-2$ - The angular momentum of the rod about the hinge remains constant throughout its fall to the floor. \(\\{\mathrm{A}\\}\) Statement \(-1\) is correct (true), Statement \(-2\) is true and Statement- 2 is correct explanation for Statement - 1 \\{B \\} Statement \(-1\) is true, statement \(-2\) is true but statement- 2 is not the correct explanation four statement \(-1\). \\{C\\} Statement \(-1\) is true, statement- 2 is false \\{D \(\\}\) Statement- 2 is false, statement \(-2\) is true

A cylinder of mass \(\mathrm{M}\) has length \(\mathrm{L}\) that is 3 times its radius what is the ratio of its moment of inertia about its own axis and that about an axis passing through its centre and perpendicular to its axis? \(\\{\mathrm{A}\\} 1\) \(\\{\mathrm{B}\\}(1 / \sqrt{3})\) \(\\{\mathrm{C}\\} \sqrt{3}\) \(\\{\mathrm{D}\\}(\sqrt{3} / 2)\)

A gramophone record of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) is rotating with angular speed \(\mathrm{W}\). If two pieces of wax each of mass \(\mathrm{M}\) are kept on it at a distance of \(R / 2\) from the centre on opposite side then the new angular velocity will be..... \(\\{\mathrm{A}\\}(\omega / 2)\) \(\\{B\\}[m \omega /(M+m)\) \(\\{C\\}[M \omega /(M+m)]\) \(\\{\mathrm{D}\\}[(\mathrm{M}+\mathrm{m}) / \mathrm{M} \omega]\)

If distance of the earth becomes three times that of the present distance from the sun then number of days in one year will be .... \(\\{\mathrm{A}\\}[365 \times 3]\) \(\\{\mathrm{B}\\}[365 \times 27]\) \(\\{\mathrm{C}\\}[365 \times(3 \sqrt{3})]\) \(\\{\mathrm{D}\\}[365 /(3 \sqrt{3})]\)
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