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Chapter 4: Problem 9

A simple pendulum is composed of a mass \(m\) attached to a massless string of length \(\ell\). The other end of the string is attached to the ceiling. A physical pendulum consists of a thin rod of uniform density with mass \(m\) and length \(\ell\). One end is attached to the ceiling by a frictionless bearing. Both the simple pendulum and the physical pendulum are displaced by an angle \(\theta\) from the vertical and released from rest. Which pendulum reaches the equilibrium \((\theta=0)\) position first? (A) The simple pendulum (B) The physical pendulum (C) They both reach the equilibrium position at the same time. (D) The answer depends on the mass of the pendulums. (E) None of the above

Short Answer

Expert verified
Answer: (B) The physical pendulum

Step by step solution

01

Formula for the period of a simple pendulum

The formula for the period of a simple pendulum is given by: \(T_s = 2\pi\sqrt{\frac{\ell}{g}}\), where \(T_s\) is the period of the simple pendulum, \(\ell\) is the length of the string, and \(g\) is the acceleration due to gravity.
02

Formula for the period of a physical pendulum

The formula for the period of a physical pendulum with a mass \(m\) and length \(\ell\) is given by: \(T_p = 2\pi\sqrt{\frac{(2/3)\ell}{g}}\), where \(T_p\) is the period of the physical pendulum.
03

Comparing the periods of the simple and physical pendulum

Comparing the expressions for the period of the simple pendulum and the physical pendulum, we see that the ratio of their periods is \(\frac{T_p}{T_s} = \frac{2\pi\sqrt{\frac{(2/3)\ell}{g}}}{2\pi\sqrt{\frac{\ell}{g}}} = \sqrt{\frac{2}{3}} < 1\). Since the ratio is less than 1, the period of the physical pendulum is shorter than the period of the simple pendulum.
04

Choosing the correct answer

We now know that the period of the physical pendulum is shorter than that of the simple pendulum. This means the physical pendulum will reach the equilibrium position first. Therefore, the correct answer is (B) The physical pendulum.

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Most popular questions from this chapter

A hollow sphere, a solid sphere, a hoop and a solid cylinder all have the same mass \(M\) and radius \(R\). Rank the following moments of inertia from highest to lowest. i. The hollow sphere spinning around a diameter ii. The solid sphere spinning around a diameter iii. The hoop spinning around an axis through its center iv. The solid cylinder spinning around an axis through its center (A) ii, iv, i, iii (B) iv, ii, i, iii (C) iii, iv, ii, i (D) i, iii, ii, iv (E) iii, i, iv, ii

A window washer of \(M=75 \mathrm{~kg}\) is taking a lunch break and sitting \(4 / 3 \mathrm{~m}\) from the left edge of a 5- meter-long board that hangs from two cables. The mass \(m\) of the board is \(20 \mathrm{~kg}\). If \(T_1\) is the tension in the right cable and \(T_2\) is the tension in the left cable, what are their values? (A) \(T_1=200 \mathrm{~N} ; T_2=650 \mathrm{~N}\) (B) \(T_1=300 \mathrm{~N} ; T_2=650 \mathrm{~N}\) (C) \(T_1=650 \mathrm{~N} ; T_2=300 \mathrm{~N}\) (D) \(T_1=650 \mathrm{~N} ; T_2=200 \mathrm{~N}\) (E) \(T_1=450 \mathrm{~N} ; T_2=400 \mathrm{~N}\)

A bullet of mass \(m\) is fired with a velocity \(v\) into a wooden disk of mass \(M\) and radius \(R\) that is free to rotate, as shown. The bullet lodges itself in the disk at a height \(b\) above the wheel's center and the system begins to rotate with an angular velocity \(\omega\). Assume \(M \gg m\). In order to predict \(\omega\), one needs (A) conservation of energy. (B) conservation of momentum. (C) conservation of angular momentum. (D) conservation of energy and conservation of momentum. (E) conservation of energy and conservation of angular momentum.

A figure skater can be idealized as composed of two solid cylinders of uniform density crossed at right angles to one another. Assume that the skater is spinning with angular speed \(\omega\) around a vertical axis passing through the center of her body. The mass of the large cylinder is \(M\); it has radius \(R\) and length \(L\). The mass of the small cylinder is \(m\); its radius is \(r\) and its length is \(\ell\). a) What is the moment of inertia of the large vertical cylinder in terms of the given quantities? b) What is the moment of inertia of the horizontal cylinder in terms of the given quantities? c) What is the figure skater's initial angular momentum? d) If \(m=0.1 M, \ell=8 R\) and the skater's arms can be totally retracted to her body, what is her new angular speed in terms of \(\omega\) ?

Which of the following could be a unit of angular momentum? i. \(\mathrm{Nm} \mathrm{S}\) ii. \(\sqrt{\mathrm{Jkg}} \mathrm{m}\) iii. \(\mathrm{W} \mathrm{s}^2\) iv. \(J / m\) v. \(W s^2 / m\) (A) \(\mathrm{i}\) (B) ii (C) iii (D) i, ii and iii (E) i, ii, iii, iv, V
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