A cart of mass is joined to a wall by a spring of stiffness and to a massless…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations and normal modes · Oscillations and normal modes

Conditions of Validity, Ch Two quadratic forms at once: the generalised eigenvalue problem

Problem

A cart of mass

m = 3\,\text{kg}

is joined to a wall by a spring of stiffness

8\,\text{N m}^{-1}

and to a massless junction (coordinate

x_2

) by a second spring of stiffness

8\,\text{N m}^{-1}

; a third spring of stiffness

8\,\text{N m}^{-1}

joins the junction to the opposite wall, so that the small-oscillation matrices are

\mathsf{M} = \begin{pmatrix}3 & 0\\0 & 0\end{pmatrix}\,\text{kg}, \qquad \mathsf{K} = \begin{pmatrix}16 & -8\\-8 & 16\end{pmatrix}\,\text{N m}^{-1},

with

x_1

the cart's displacement and

x_2

the junction's displacement. ((a)) Explain why the second row of

\mathsf{K}\mathbf{x} = -\mathsf{M}\ddot{\mathbf{x}}

is a static constraint rather than an equation of motion, and solve it for

x_2

in terms of

x_1

. ((b)) Eliminate

x_2

and find the single surviving angular frequency

\omega