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Chapter 11: Problem 6

Three identical springs, of negligible mass, spring constant \(k\), and natural length \(a\) are attached end-to-end, and a pair of particles, each of mass \(m\), are fixed to the points where they meet. The system is stretched between fixed points a distance \(3 l\) apart \((l>a)\). Find the frequencies of normal modes of (a) longitudinal, and (b) transverse oscillations.

Short Answer

Expert verified
#tag_title# Step 3: Solve for the frequencies of the normal modes #tag_content# To find the frequencies of the normal modes, we need to solve the equations of motion for both particles as a system of linear differential equations. This can be done using either the Newtonian approach or the Lagrangian or Hamiltonian methods. Once the normal modes are found, the frequencies can be determined from the characteristic equations.

Step by step solution

01

Visualize the problem

Draw a diagram of the system showing three identical springs connected end-to-end, with two particles fixed at the points where they meet, and the entire system stretched between fixed points 3l apart.
02

Write the equations of motion for each particle

Write the equations of motion for both particles in the longitudinal direction, considering the forces acting on each particle. In the case of longitudinal oscillations, use Hooke's law for the spring forces. For transverse oscillations, consider the forces due to the tension in the springs and the restoring forces due to the spring force.

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Most popular questions from this chapter

A rigid rod of length \(2 a\) is suspended by two light, inextensible strings of length \(l\) joining its ends to supports also a distance \(2 a\) apart and level with each other. Using the longitudinal displacement \(x\) of the centre of the rod, and the transverse displacements \(y_{1}, y_{2}\) of its ends, as generalized co-ordinates, find the Lagrangian function (for small \(\left.x, y_{1}, y_{2}\right)\). Determine the normal modes and frequencies. (Hint: First find the height by which each end is raised, the co-ordinates of the centre of mass and the angle through which the rod is turned.)

A particle of charge \(q\) and mass \(m\) is free to slide on a smooth horizontal table. Two fixed charges \(q\) are placed at \(\pm a \boldsymbol{j}\), and two fixed charges \(12 q\) at \(\pm 2 a i\). Find the electrostatic potential near the origin (see \(\S 6.2\) ). Show that this is a position of stable equilibrium, and find the frequencies of the normal modes of oscillation near it.

A spring of negligible mass, and spring constant (force/extension) \(k\), supports a mass \(m\), and beneath it a second, identical spring, carrying a second, identical mass. Using the vertical displacements \(x\) and \(y\) of the masses from their positions with the springs unextended as generalized co- ordinates, write down the Lagrangian function. Find the position of equilibrium, and the normal modes and frequencies of vertical oscillations.

A particle moves under a conservative force with potential energy \(V(\boldsymbol{r})\). The point \(\boldsymbol{r}=\mathbf{0}\) is a position of equilibrium, and the axes are so chosen that \(x, y, z\) are normal co- ordinates. Show that, if \(V\) satisfies Laplace's equation, \(\boldsymbol{\nabla}^{2} V=0\) (see \(\left.\S 6.7\right)\), then the equilibrium is necessarily unstable, and hence that stable equilibrium under purely gravitational and electrostatic forces is impossible. (Of course, dynamic equilibrium stable periodic motion - can occur. Note also that the two- dimensional stable equilibrium of Problem 11 does not contradict this result because there is another force imposed, confining the charge to the horizontal plane.)

A simple pendulum of mass \(m\), whose period when suspended from a rigid support is \(1 \mathrm{~s}\), hangs from a supporting block of mass \(2 m\) which can move along a horizontal line (in the plane of the pendulum), and is restricted by a harmonic-oscillator restoring force. The period of the oscillator (with the pendulum removed) is \(0.1 \mathrm{~s}\). Find the periods of the two normal modes. When the pendulum bob is swinging in the slower mode with amplitude \(100 \mathrm{~mm}\), what is the amplitude of the motion of the supporting block?
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