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Chapter 4: Problem 455

What is the velocity of the bob of a simple pendulum at its mean position, if it is able to rise to vertical height of \(18 \mathrm{~cm}\) (Take \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\) ) (A) \(0.4 \mathrm{~m} / \mathrm{s}\) (B) \(4 \mathrm{~m} / \mathrm{s}\) (C) \(1.8 \mathrm{~m} / \mathrm{s}\) (D) \(0.6 \mathrm{~m} / \mathrm{s}\)

Short Answer

Expert verified
The correct answer is not in the given options. When applying the principle of energy conservation for the given simple pendulum problem, the velocity of the pendulum at its mean position is found to be \(6 \mathrm{~m} / \mathrm{s}\).

Step by step solution

01

Understand the Problem

The problem is asking for the velocity of a pendulum bob at its mean position, given the height it achieves when it swings. We will use the principle of conservation of energy to solve it. No matter is given for the bob, so we can ignore it during our calculations because it will be cancelled out.
02

Potential Energy at Maximum Height

The bob's potential energy at its maximum height is given by the formula \(PE = mgh\), where \(PE\) is the potential energy at maximum height, \(m\) is mass of the bob (which we don't need), \(g\) is the acceleration due to gravity and \(h\) is the height. Substituting the given values: \(PE = m * 10 * 0.18\). We don't know the mass of the bob, but we know that it will be the same in our next calculation, so we can keep the formula as it is.
03

Kinetic Energy at Mean Position

The bob's kinetic energy at its mean position is given by the formula \(KE = 0.5 * m * v^2\), where \(KE\) is the kinetic energy, \(m\) is mass (which we don't need), and \(v\) is velocity (which we're trying to find). Since the potential energy at the height is completely converted into kinetic energy at the pendulum's mean position, we can say that \(PE = KE\).
04

Solving for Velocity

Equating the expressions for potential and kinetic energy derived in Step 2 and Step 3 gives us: \(m * 10 * 0.18 = 0.5 * m * v^2\). Notice that 'm' will cancel out. Solving the equation for 'v' (velocity), which is the problem's objective, gives us \(v = \sqrt{2 * 10 * 0.18}\). As a result, the velocity of the bob at the mean position is equal to \(v = 6 m/s\).
05

Conclusion

None of the provided options is correct. The correct solution is \(6 m/s\), which resulted from applying the principle of energy conservation to a simple pendulum problem.

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Most popular questions from this chapter

A body having a mass of \(0.5 \mathrm{~kg}\) slips along the wall of a semispherical smooth surface of radius \(20 \mathrm{~cm}\) shown in figure. What is the velocity of body at the bottom of the surface $?\left(\mathrm{~g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$ (A) \(2 \mathrm{~m} / \mathrm{s}\) (B) \(2 \mathrm{~m} / \mathrm{s}\) (C) \(2 \sqrt{2} \mathrm{~m} / \mathrm{s}\) (D) \(4 \mathrm{~m} / \mathrm{s}\)

A particle is acted upon by a force \(\mathrm{F}\) which varies with position \(\mathrm{x}\) as shown in figure. If the particle at \(\mathrm{x}=0\) has kinetic energy of \(20 \mathrm{~J}\). Then the calculate the kinetic energy of the particle at \(\mathrm{x}=16 \mathrm{~cm}\). (A) \(45 \mathrm{~J}\) (B) \(30 \mathrm{~J}\) (C) \(70 \mathrm{~J}\) (D) \(135 \mathrm{~J}\)

The velocity of a body of mass \(400 \mathrm{gm}\) is $(-3 \mathrm{i} \wedge-4 \mathrm{j} \wedge) \mathrm{m} / \mathrm{s}$. So its kinetic energy is ...... (A) \(5 \mathrm{~J}\) (B) \(10 \mathrm{~J}\) (C) \(8 \mathrm{~J}\) (D) \(16 \mathrm{~J}\)

A rifle bullet loses \((1 / 10)^{\text {th }}\) of its velocity in passing through a plank. The least number of such planks required just to stop the bullet is (A) 5 (B) 10 (C) 11 (D) 20

If a skater of weight \(4 \mathrm{~kg}\) has initial speed $4 \mathrm{~m} / \mathrm{s}\( and \)2^{\text {nd }}\( one of weight \)6 \mathrm{~kg}\( has \)6 \mathrm{~m} / \mathrm{s}\(. After collision, they have speed (couple) \)6 \mathrm{~m} / \mathrm{s}\(. Then the loss in \)\mathrm{K} . \mathrm{E}$. is..... (A) \(48 \mathrm{~J}\) (B) zero (C) \(96 \mathrm{~J}\) (D) None of these
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