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Mechanics Berkeley Physics

Charles Kittel, Walter D. Knight, Malvin A. Ruderman, A. Carl Helmholz, Burton J. Moyer

Chapter 7

Harmonic Oscillator: Properties and Examples - all with Video Answers

Educators

Chapter Questions

02:04

Problem 1

Simple pendulum. A pendulum is constructed from a light string of length $L=100 \mathrm{~cm}$ and a heavy mass $M=1 \times 10^{3} \mathrm{~g}$. What is the period of the pendulum for small displacements?

Urvashi Arora
Urvashi Arora
Numerade Educator
00:43

Problem 2

Mass on a spring. Write the equation of motion for a mass $M$ moving in a vertical line under the action of gravity and a spring of spring constant $C .$ What is the effect of gravity on:
(a) The period of oscillation?
(b) The center of oscillation, the point about which the oscillation occurs?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:19

Problem 3

Mass on a spring. A mass of $1.0 \times 10^{3} \mathrm{~g}$ is suspended from a linear spring with a spring constant $C=1.0 \times 10^{6} \mathrm{dyn} / \mathrm{cm}$.
(a) What is the period for small oscillations?
(b) If at $t=0$ the displacement from equilibrium is $+0.5 \mathrm{~cm}$ and the velocity is $+15 \mathrm{~cm} / \mathrm{s}$, find the displacement as a function of $t$.

Lucas Finney
Lucas Finney
Numerade Educator
02:10

Problem 4

Mass on a spring-data. The data in Tables $7.3$ and $7.4$ were obtained by observing the motion of a mass attached to the end of a spring:
(a) Plot the square of the period of oscillation as a function of mass. The tabulated values of mass exclude the mass of the spring. Determine the effective mass of the spring by proper extrapolation of the graph.
(b) Determine the spring constant $C$.
(c) Plot the natural logarithm of the amplitude as a function of time and determine the relaxation time.
$(d)$ Determine the damping factor $b$

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
03:26

Problem 5

Mass in buoyant medium. A body partly (or completely) submerged in liquid is buoyed up by a force equal to the weight of the liquid displaced (Archimedes' principle). Show that a body of uniform horizontal cross section constrained to move vertically in a liquid of density greater than the density of the body will execute simple harmonic oscillations. What is the period? What is the limit of amplitude of the oscillations?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
04:19

Problem 6

Pendulum
(a) A pendulum of length $39.2 \mathrm{~cm}$ and mass $500 \mathrm{~g}$ is set in motion so that at $t=0, \theta=0.1$, and $\dot{\theta}=-0.02 / \mathrm{s}$. Find $\theta$ as a function of $t$. Now use the equations of motion to find the force on the mass at $\theta=0$.
(b) The Foucault pendulum was set up by Foucault in 1851 in Paris to show the effect of the rotation of the earth (see Chap. 4, page 114$)$. Its length is $69 \mathrm{~m}$. Find the period. If the mass is $28 \mathrm{~kg}$ and the maximum swing is $10^{\circ}$, find the total energy of motion.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:57

Problem 7

Energy of mass on spring. A mass of $50 \mathrm{~g}$ on the end of a certain spring executes simple harmonic motion according to the equation $x=2 \sin 10 t$ where $x$ is in centimeters and $t$ is in seconds.
(a) Find the spring constant $C$.
(b) Find the maximum kinetic energy.
(c) What are the maximum potential energy and the total energy?

Urvashi Arora
Urvashi Arora
Numerade Educator
02:15

Problem 8

Two-dimensional oscillator. A particle is free to move in the $x, y$ plane under the action of a force toward the origin of magnitude $-C(x \hat{\mathrm{x}}+y \hat{y})=-C r$. Assuming the mass is $M$, find the $x$ and $y$ equations of motion and solve them.
(a) What are the conditions for motion in a circle and what is the period?
(b) What are the conditions for motion along the line at $45^{\circ}$ to the $x$ axis and what is the period?

Manish Jain
Manish Jain
Numerade Educator
02:25

Problem 9

Mass in spherical bowl. A mass slides freely in the bottom of a spherical bowl of radius $1,0 \mathrm{~m}$. Find the period for small oscillations. What is the length of the equivalent pendulum?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
02:28

Problem 10

Viscosity
(a) Consult a reference book to give and explain the definition of viscosity.
(b) What are the dimensions of the coefficient of viscosity $\eta ?$
(c) What is the value of the viscosity of water at $20^{\circ} \mathrm{C} ?$
(d) Evaluate $(b)$ [see Eq. $(7.28)]$ for a sphere of radius $5 \mathrm{~cm}$ in a medium of viscosity $2.0$ centipoises. (See Prob. 13.)
(e) If in $(d)$ the density of the sphere is $2.7 \mathrm{~g} / \mathrm{cc}$ and the density of the liquid is $1.1 \mathrm{~g} / \mathrm{cc}$, find the terminal velocity. Use the net vertical force, as explained in Prob. 5, and the relaxation time.

Averell Hause
Averell Hause
Carnegie Mellon University
02:48

Problem 11

Motion under a viscous force
(a) A particle of mass $M$ acted on by only the viscous force of the medium $-b v$ is projected from a point with velocity $v_{0}$. Write down the velocity as a function of time. Remembering that $v=d x / d t$, find $x(t) .$ If $M=10 \mathrm{~g}$ $b=4.0 \mathrm{dyn}-\mathrm{s} / \mathrm{cm}$, and $v_{0}=100 \mathrm{~cm} / \mathrm{s}$, find the distance
the mass will travel.
(b) In the Millikan oil-drop experiment some drops had radius $2.0 \times 10^{-4} \mathrm{~cm} .$ The density of the oil was $0.92 \mathrm{~g} / \mathrm{cc}$, and the viscosity of air was $1.8 \times 10^{-2}$ centipoises. Find the relaxation time and the terminal velocity. Neglect the buoyant correction.

Ajay Singhal
Ajay Singhal
Numerade Educator
02:58

Problem 12

Relaxation time. An oscillator has $M=10 \mathrm{~g}$, $C=490 \mathrm{dyn} / \mathrm{cm}, \quad b=1.0 \mathrm{dyn}-\mathrm{s} / \mathrm{cm} .$ At $t=0, x=2.0 \mathrm{~cm}$
$\dot{x}=0:$
(a) Find $x$ as a function of $t$.
(b) What is the relaxation time for $x$; for $K$ ?
(c) What is the $Q ?$

Shoukat Ali
Shoukat Ali
Other Schools
02:14

Problem 13

13. Damped oscillator. A spherical ball of radius $0.30 \mathrm{~cm}$ and mass $0.5 \mathrm{~g}$ moves in water under the action of a spring of constant $C=50 \mathrm{dyn} / \mathrm{cm} . \eta$ for water is $1.0 \times 10^{-2} \mathrm{dyn}-\mathrm{s} / \mathrm{cm}^{2}$
or poises. Find the number of oscillations that will occur in the time for the amplitude to drop to one-half the initial amplitude. (Note that $e^{-0.693}=\frac{1}{2}$.) What is the $Q$ of the oscillator?

Prashant Bana
Prashant Bana
Numerade Educator