Chapter 14

Imagine you are an astronaut who has landed on another planet and wants to determine the free-fall acceleration on that planet. In one of the experiments you decide to conduct, you use a pendulum $0.50\mathrm{m}$ long and find that the period of oscillation for this pendulum is $1.50\mathrm{s}$. What is the acceleration due to gravity on that planet?

A horizontal tree branch is directly above another horizontal tree branch. The elevation of the higher branch is $9.65\mathrm{m}$ above the ground, and the elevation of the lower branch is $5.99\mathrm{m}$ above the ground. Some children decide to use the two branches to hold a tire swing. One end of the tire swing's rope is tied to the higher tree branch so that the bottom of the tire swing is $0.47\mathrm{m}$ above the ground. This swing is thus a restricted pendulum. Start. ing with the complete length of the rope at an initial angle of ${14.2}^{\circ }$ with respect to the vertical, how long does it take a child of mass $29.9\mathrm{kg}$ to complete one swing back and forth?

Two pendulums of identical length of $1.000\mathrm{m}$ are suspended from the ceiling and begin swinging at the same time. One is at Manila, in the Philippines, where $g=9.784\mathrm{m}/{\mathrm{s}}^2$ and the other is at Oslo, Norway, where $g=9.819\mathrm{m}/{\mathrm{s}}^2,$ After how many oscillations of the Manila pendulum will the two pendulums be in phase again? How long will it take for them to be in phase again?

Two springs, each with $k=125\mathrm{N}/\mathrm{m}$, are hung vertically, and $1.00-\mathrm{kg}$ masses are attached to their ends. One spring is pulled down $5.00\mathrm{cm}$ and released at $t=0$; the othen is pulled down $4.00\mathrm{cm}$ and released at $t=0.300\mathrm{s}$. Find the phase difference, in degrees, between the oscillations of the two masses and the equations for the vertical displacements of the masses, taking upward to be the positive direction.

The period of a pendulum is $0.24\mathrm{s}$ on Earth. The period of the same pendulum is found to be 0.48 s on planet $X,$ whose mass is equal to that of Earth. (a) Calculate the gravitational acceleration at the surface of planet $X$. (b) Find the radius of planet $\mathrm{X}$ in terms of that of Earth.

A grandfather clock uses a pendulum and a weight. The pendulum has a period of $2.00\mathrm{s}$, and the mass of the bob is 250. $\mathrm{g}$. The weight slowly falls, providing the energy to overcome the damping of the pendulum due to friction. The weight has a mass of $1.00\mathrm{kg}$, and it moves down $25.0\mathrm{cm}$ every day. Find $Q$ for this clock. Assume that the amplitude of the oscillation of the pendulum is ${10.0}^{\circ }$

A cylindrical can of diameter $10.0\mathrm{cm}$ contains some ballast so that it floats vertically in water. The mass of can and ballast is $800.0\mathrm{g}$, and the density of water is $1.00\mathrm{g}/{\mathrm{cm}}^3$ The can is lifted $1.00\mathrm{cm}$ from its equilibrium position and released at $t=0.$ Find its vertical displacement from equilibrium as a function of time. Determine the period of the motion. Ignore the damping effect due to the viscosity of the water.

The period of oscillation of an object in a frictionless tunnel running through the center of the Moon is $T=2\pi /{\omega }_0$ $=6485\mathrm{s}$, as shown in Fxample 142 . What is the period of oscillation of an object in a similar tunnel through the Earth $(R_{\mathrm{I}}=6.37\cdot {10}^6\mathrm{m};R_{\mathrm{M}}=1.74\cdot {10}^6\mathrm{m};M_{\mathrm{E}}=5.98\cdot {10}^{24}\mathrm{kg}$ $M_u=7.36\cdot {10}^{22}\mathbf{k}\mathbf{g})?$

The motion of a planet in a circular orbit about a star obeys the equations of simple harmonic motion. If the orbit is observed edge-on, so the planet's motion appears to be onedimensional, the analogy is quite direct: The motion of the planet looks just like the motion of an object on a spring a) Use Kepler's Third Law of planetary motion to determine the "spring constant" for a planet in circular orbit around a star with period $T$ b) When the planet is at the extremes of its motion observed edge-on, the analogous "spring" is extended to its largest displacement. Using the "spring" analogy, determine the orbital velocity of the planet.

An object in simple harmonic motion is isochronous, meaning that the period of its oscillations is independent of their amplitude. (Contrary to a common assertion, the operation of a pendulum clock is not based on this principle. A pendulum clock operates at fixed, finite amplitude. The gearing of the clock compensates for the anharmonicity of the pendulum.) Consider an oscillator of mass $m$ in one-dimensional motion, with a restoring force $F(x)=-c{x}^3$ where $x$ is the displacement from equilibrium and $c$ is a constant with appropriate units. The motion of this ascillator is periodic but not isochronous. a) Write an expression for the period of the undamped oscillations of this oscillator. If your expression involves an integral, it should be a definite integral. You do not need to evaluate the expression. b) Using the expression of part (a), determine the dependence of the period of oscillation on the amplitude. c) Generalize the results of parts (a) and (b) to an oscillator of mass $m$ in one-dimensional motion with a restoring force corresponding to the potential energy $U(x)=\gamma |x|/\alpha$, where $\alpha$ is any positive value and $\gamma$ is a constant