What is Small oscillations: Definition and 63 Discussions
The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:
sin
θ
≈
θ
cos
θ
≈
1
−
θ
2
2
≈
1
tan
θ
≈
θ
{\displaystyle {\begin{aligned}\sin \theta &\approx \theta \\\cos \theta &\approx 1-{\frac {\theta ^{2}}{2}}\approx 1\\\tan \theta &\approx \theta \end{aligned}}}
These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science. One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision.
There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation,
cos
θ
{\displaystyle \textstyle \cos \theta }
is approximated as either
[Moved from technical forums, so no template]
Summary:: A rod of length l and mass m, pivoted at one end, is held by a spring at its midpoint and a spring at its far end, both pulling in opposite directions. The springs have spring constant k, and at equilibrium their pull is perpendicular to...
Please note that the transformed quantities will be indicated by ##'##.
Let me give some context first.
Let us assume here that the general approximate form of the potential energy ##V## and the kinetic energy ##T## are given to be
$$V^{app} = q^T V q \tag 1$$
$$T^{app} = \dot q^T V \dot q...
Hi,
Given a mechanic-problem, I've linearised a system of two differential equations, which the origin was Lagrange-equations.
The system looks like this;
$$ 5r \ddot{\theta} + r \ddot{\phi} + 4g \theta = 0´ \\ 3r \ddot{\theta} + 2r \ddot{\phi} + 3g \phi = 0 $$
$$ $$
And I shall find the...
So I'm working on this home assignment that has numerous segments. Firstly, I was asked to find the equilibrium distance between two particles in a potential well described by U(r).
I did that by setting U'(r) = 0 and came out with r_equilibrium = 2^(1/6)*a.
Now, I'm being asked to find the...
Hey, I solved a problem about a double pendulum and got 2 euler-lagrange equations:
1) x''+y''+g/r*x=0
2) x''+y'' +g/r*y=0 (where x is actually a tetha and y=phi)
the '' stand for the 2nd derivation after t, so you can see the basic harmonic oscillator equation with a term x'' or y'' that...
I first found the equilibrium points taking the derivative of the potential. ##U'(x)=U_0 a\sin(ax)##, and the equilibrum is when the derivative is 0, so ##U_0 a\sin(ax)=0## so ##x=0## or ##x=\pi/a##. Taking the second derivative ##U''(x)=U_0a^2 \cos(ax)## I find that ##x=0## is a minimum point...
I tried by taking the derivative of the potential to find the critic points and the I took the second derivative to find which of those points are minimum points. I found that the point is ##x=- a##. I don't understand how to calculate the period, since I haven't seen anything about the harmonic...
Homework Statement
A uniform rod of mass M, and length L swings as a pendulum with two horizontal springs of negligible mass and constants k1 and k2 at the bottom end as shown in the figure. Both springs are relaxed when the when the rod is vertical. What is the period T of small oscillations...
Homework Statement
The problem is taken from Morin's book on classical mechanics. I found out Lagrangian of motion. Now to solve, we need small angle and small x approximation. The small angle approximation is easy to treat. But how to solve small x approximation i.e how do I apply it...
Homework Statement
One silly thing may be I am missing for small oscillations of a pendulum the potential energy is -mglcosθ ,for θ=0 is the point of stable equilibrium (e.g minimum potential energy) .Homework Equations
Small oscillations angular frequency
ω=√(d2Veffect./mdθ2) about stable...
Homework Statement
Two balls of mass m are attached to ends of two, weigthless metal rods (lengths l1 and l2). They are connected by another metal bar.
Determine period of small oscillations of the system
Homework Equations
Ek=mv2/2
v=dx/dt
Conversation of energy
2πsqrt(M/k)
The Attempt at a...
Can anyone tell me how if the derivative of n(n') is quadratic the second term in the taylor series expansion given below vanishes. This doubt is from the book Classical Mechanics by Goldstein Chapter 6 page 240 3rd edition. I have attached a screenshot below
Homework Statement
[/B]
Here's the problem from the homework. I've called the initial positions in order as 0, l, and 2l.
Homework Equations
The most important equation here would have to be
|V - w2*M| = 0,
where V is the matrix detailing the potential of the system and M as the "masses" of...
Homework Statement
A point mass m is fixed inside a hollow cylinder of radius R, mass M and moment of inertia I = MR^2. The cylinder rolls without slipping
i) express the position (x2, y2) of the point mass in terms of the cylinders centre x. Choose x = 0 to be when the point mass is at the...
Homework Statement
In an infinite flat layer of thickness 2d, volume charge density is given according to the law: ρ=(ρ°)(x)/d and (-d≤x≤d). Here, x is the axis perpendicular to the plane. In the layer, there is a thin channel in which a point dipole of mass m and dipole moment p is placed...
Homework Statement
The potential energy of a particle of mass m is V(r) = k/r + c/3r^3 where k<0 and c is a small constant. Find the angular velocity \omega in a circular orbit of radius a and the angular frequency \omega' of small radial oscillations about this circular orbit. Hence show...
Two bodies of mass m each are attached by a spring. This two body system rotates around a large mass M under gravity. Will there be any relation between frequency of oscillation of the two body system and frequency of rotation?
Frequency of small oscillations of a single body rotating in an...
Homework Statement A pendulum consists of a light rigid rod of length 250 mm, with two identical uniform solid spheres of radius of radius 50 mm attached one on either side of its lower end. Find the period of small oscillations (a) perpendicular to the line of centres and (b) along...
Homework Statement
"Take a PE function U(x), which has an equilibrium point at x=0, and provides a restoring force in that region, and show that a Taylor expansion around that area can be approximated by a SHO PE function for small x."
Homework Equations
U=.5kx^2...x =...
Homework Statement
The potential energy of a particle of mass m near the position of equilibrium is given by U=U0sin2(αx) where U0 and α are constants. Find the frequency of the small oscillations about the position of equilibrium.
Homework Equations
Work energy equation...
Problem from past PGRE:
A particle of mass m moves in a one-dimensional potential V(x)=-ax2 + bx4, where a and b are positive constants. The angular frequency of small oscillations about the minima of the potential is equal to:
Answer is 2(a/m)1/2.
I understand how this is found 'the long...
Homework Statement
See picture :
Homework Equations
##\sum M_{O}=I_{O}\ddot{\theta }##
The Attempt at a Solution
Consider the free-body diagram associated with an arbitrary positive angular displacement ##\theta##; The moment about point ##O## is given by
##\sum...
Homework Statement
A small ball of radius r performes small oscillations within a hollow cylinder of radius R. What would be the angular frequency of the oscillations given that the rolling is without slipping? The angle between the radius connecting the center of the hollow cylinder to the...
Q: http://gyazo.com/1ee7eee0134c25a23b4ad7a6972e1e46
part a)
I have drawn the graph and calculated ## V'(x) = \dfrac{3\lambda x^2 (x^4 + a^4) - \lambda x^3(4x^3)}{(x^4+a^4)^2} = 0 ## and found using the graph that the value of x when the particle is in a stable equilibrium is ## x=...
Homework Statement
A spring of rest length L (no tension) is connected to a support at one of the ends and has a mass m attached to the other. Write the Lagrange equations. Discuss the movement for small departures from equilibrium.
Homework Equations
I did the first part, and I found...
Homework Statement
From Goldstein Classical Mechanics, 6.16:
A mass particle moves in a constant vertical gravitational field along the curve defined by y=ax4 , where y is the vertical direction. Find the equation of motion for small oscillations about the position of equilibrium.
The...
Homework Statement
I'm studying small oscillations. When can I say that an orbit is closed?
The Attempt at a Solution
I remember that there is a ratio that must be a rational number but I don't remember other thing...
Thank you!
Homework Statement
I'm solving an exercise about small oscillations.
I name T the kinetic matrix and $H$ the hessian matrix of potential.
The matrix \omega^2 T- H is diagonal and so find the auto-frequencies is easy! But I have a problem with normal modes. The lagrangian coordinates are two...
Hi,
I'm studying Small Oscillations and I'm having a problem with normal modes.
In some texts, there is written that normal modes are the eigenvectors of the matrix $V- \omega^2 V$ where V is the matrix of potential energy and T is the matrix of kinetic energy.
Some of them normalize the...
Dear friends,
In small oscillation problem of classical mechanics, the generalized force is zero (Eq. (6-1)) of the Goldstein book.
The suffix 'i' represent the the particle number or coordinate?
In Eq. (6-3) of the same problem the summation convention is used. what is the purpose of...
Homework Statement
Basically the issue is Landau & Lifgarbagez mechanics says
δl = [r2 + (l + r)2 - 2r(l + r)cosθ]1/2 - l ≈ r(l + r)θ2/2l
Homework Equations
θ much less than 1
The Attempt at a Solution
I've no idea how to get the thing on the far right. I'm assuming it's...
Homework Statement
v(x)= (1/x^2) -(1/x) Find the frequency of small osciallations about the equilibrium point
Homework Equations
The Attempt at a Solution
I have so far worked out the equilibrium point is at x=2, to get this i differentiated v(x) and solved it, but could...
I'm not sure where to post this question. In classical mechanics many problems are simplified in the approximation of "small angles" or "small oscillations".
Wikipedia gives the following criteria or approximations:
\sin \theta \approx \theta.
\cos \theta \approx 1 - \frac{\theta ^2}{2}
\tan...
Homework Statement
Consider 2 masses linked via 3 springs in this way |----m----m----| where the | denotes fixed walls and the ---- the springs.
The length between the walls is 2L and the natural length of each spring is b=L/3.
When we move a mass from its equilibrium position, each spring...
Homework Statement
I'm stuck at understanding how to find the kinetic and potential energy matrices such that the determinant |V- \omega ^2 T|=0 when solved for \omega, gives the normal modes (characteristic frequencies?) of the considered system.
For example in Goldstein's book for a molecule...
Homework Statement
I'm trying to teach myself Small Oscillations in Classical Mechanics. So far I've read in Landau, Golstein, Wikipedia and other internet sources but this subjet seems really tough to even understand to me.
What I understand is that if we have a potential function that...
Homework Statement
The first part of the problem is just finding the Lagrangian for a system with 2 d.o.f. and using small angle approximations to get the Lagrangian in canonical/quadratic form, not a problem. I am given numerical values for mass, spring constants, etc. and am told to find the...
Hi guys i am a bit confused about this problem,
a particle of mass, m, moves in potential a potential u(x)=k(x4 - 7 x2 -4x)
I need to find the frequency of small oscillations about the equilibrium point.
I have worked out that x=2 corresponds to the equilibrium point as
- dU/dx = F =...
Homework Statement
Consider a system of one generalized coordinate theta, having the following Lagrangian equation of motion:
r and b are constants
m is mass
(1/3)mb^{2}\ddot{\theta} = r(r+b)\theta + r^{2}\theta^{3} + gr\theta
And this potential energy (if it matters):
U = mg(r+b)...
1. A uniform coin with radius R is pivoted at a point that is a distance d from its center. The coin is free to swing back and forth in the vertical plane defined by the plane of the coin. For what value of d is the frequency of small oscillations largest?
2. V(x)\equivpotential energy...
Homework Statement
Find the frequency of oscillations of a particle (mass m) which is free to move along the parabola y= -ax^2 + 2ax - a, and is attached to an ideal spring whose other end is fixed at (1,l) A force F is required to extend the spring to length l. a can be any real number...
Homework Statement
A rod of length L and mass m, pivoted at one end, is held by a spring at its midpoint and a spring at its far end, both pulling in opposite directions. The springs have spring constant k, and at equilibrium their pull is perpendicular to the rod. Find the frequency of small...
A point of mass slides without friction on a horizontal table at one end of a massless spring of natural length a and spring const k as shown in the figure below. The other end of the spring is attached to the table so it can rotate freely without friction. The spring is driven by a motor...
Homework Statement
Consider an atom of mass m bonded to the surface of a much larger immobile body by
electromagnetic forces. The force binding the atom to the surface has the expression
F = eacosz + bsinz + dtanz
where a, b and d are constants and z is positive upwards. The...
A horizontal arrangement with 1 spring in between the two masses, 1 spring connecting each mass to opposite fixed points:
k 3m k 8m k
|----[]----[]----|
I solved the eigenvalue/eigenvector problem for the dynamical matrix D where V = 1/2 D_{ij} w_i w_j and the w's are...
Homework Statement
Consider two pendulums, I and II. I consists of a bob of mass 2m at the end of a rod of length L. II consists of one bob of mass m at the end of a rod of length L and another bob of mass m halfway up the road, at L/2. What is the ratio of the frequency of small...
A particle of mass m moves in one dimension subject to the potential:
V(x)=(-12/x)+(x^-12)
Find the equilibrium point and the frequency of small oscillations about that point.
I think I've found the equilibrium point 'a', but using the formula V'(a)=0, and i got the answer a=1...
Homework Statement
The problem is: A point pendulum is being accelerated at a constant acceleration of a. Basically what's required is to find the equations of motion, the equilibrium point, and to show that the frequency of small oscillations about the e.p. is: \omega=L^{-1/2}...
Homework Statement
A particle of mass m and charge q can move along a vertical circle of radius R in the constant gravitational field of the earth. Another charge q is fixed to the lowest point of teh circle.
Find the equilibrium position and the frequency of small oscillations of the...