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Chapter 7: Problem 26

If the length of a simple pendulum is equal to the radius \(R\) of the earth, its time period will be (A) \(2 \pi \sqrt{R / g}\) (B) \(2 \pi \sqrt{R / 2 g}\) (C) \(2 \pi \sqrt{2 R / g}\) (D) \(\pi \sqrt{R / 2 g}\)

Short Answer

Expert verified
The correct answer for the time period of a simple pendulum with length equal to the radius of the Earth is option (C) \(2 \pi \sqrt{2 R / g}\).

Step by step solution

01

Recall the time period formula for a simple pendulum

The time period \(T\) of a simple pendulum is given by the formula: \[ T = 2 \pi \sqrt{\frac{L}{g}} \] where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity. In this exercise, the length of the pendulum is equal to the radius of Earth \(R\).
02

Derive the altitude-dependent gravity

At the surface of the Earth, the acceleration due to gravity is \(g = 9.81 m/s^2\). However, gravity weakens as you go away from the Earth's surface. The altitude-dependent gravity can be expressed as: \[ g_a = \frac{GM}{(R+a)^2} \] where \(g_a\) is the acceleration due to gravity at an altitude \(a\), \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(a\) is the altitude.
03

Calculate the gravity at an altitude equal to the Earth's radius

In this case, we need to find the gravity at an altitude \(a = R\). So, the gravity at this altitude is: \[ g_a = \frac{GM}{(R+R)^2} = \frac{GM}{(2R)^2} \]
04

Substitute the values in the time period formula

Now, we can substitute the length \(L=R\) and the gravity \(g=g_a\) in the time period formula: \[ T = 2\pi\sqrt{\frac{R}{\frac{GM}{(2R)^2}}} \]
05

Simplify the formula to find the correct option

After substituting the values and simplifying the equation, we get the formula for the time period of the pendulum: \[ T = 2 \pi \sqrt{\frac{2R^3}{GM}} \] This formula corresponds to the option (C) because it has the same form, but we have to divide both numerator and denominator by \(R\): \[ T = 2 \pi \sqrt{\frac{2 R}{g}} \] Therefore, the correct answer is option (C) \(2 \pi \sqrt{2 R / g}\).

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Most popular questions from this chapter

The orbital velocity of an artificial satellite in a circular orbit just above earth's surface is \(v_{0} .\) For a satellite orbiting in a circular orbit at an altitude of half of earth's radius is (A) \(\sqrt{\frac{3}{2}} v_{0}\) (B) \(\sqrt{\frac{2}{3}} v_{0}\) (C) \(\frac{3}{2} v_{0}\) (D) \(\frac{2}{3} v_{0}\)

In 1783, John Mitchell noted that if a body having same density as that of the sun but radius 500 times that of the sun, magnitude of its escape velocity will be greater than \(c\), the speed of light. All the light emitted by such a body will return to it. He, thus, suggested the existence of a black hole. \(v=c=\sqrt{\frac{2 G M}{R}}\) suggests that a body of mass \(M\) will act as a black hole if its radius \(R\) is less than or equal to a certain critical radius. Karl Schwarzchild, in 1926 , derived the expression for the critical radius \(R_{S}\) called Schwarzchild radius. The surface of the sphere with radius \(R_{S}\) surrounding a black hole is called event horizon. To make black hole with density of the sun, the ratio of radius of the object to that of sun should be (A) 5 (B) 50 (C) 500 (D) \(2.5\)

The period of revolution of planet \(A\) around the sun is 8 times that of \(B\). The distance of \(A\) from the sun is how many times greater than that of \(B\) from the sun? (A) 2 (B) 3 (C) 4 (D) 5

If three uniform spheres, each having mass \(M\) and radius \(R\), are kept in such a way that each touches the other two, the magnitude of the gravitational force on any sphere due to the other two is (A) \(\frac{G M^{2}}{4 R^{2}}\) (B) \(\frac{2 G M^{2}}{R^{2}}\) (C) \(\frac{2 G M^{2}}{4 R^{2}}\) (D) \(\frac{\sqrt{3} G M^{2}}{4 R^{2}}\)

A body of mass \(m\) rises to a height \(h=R / 5\) from the earth's surface where \(R\) is radius of the earth. If \(g\) is acceleration due to gravity at the earth surface, the increase in potential energy is (A) \(m g R\) (B) \((4 / 5) m g R\) (C) \((1 / 6) m g R\) (D) \((6 / 7) m g R\)
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