A block of mass hangs from a massless rigid plate (coordinate for the plate

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 block of mass

m = 3\,\text{kg}

hangs from a massless rigid plate (coordinate

x_1

for the plate,

x_2

for the block), all motion vertical about equilibrium. The ceiling is joined to the plate by a spring of stiffness

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

, the plate to the block by a spring of stiffness

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

, and the block to the fixed floor by a spring of stiffness

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

, so that

\mathsf{M} = \begin{pmatrix}0&0\\0&3\end{pmatrix}\,\text{kg}, \qquad \mathsf{K} = \begin{pmatrix}24&-12\\-12&24\end{pmatrix}\,\text{N m}^{-1}.

((a)) Write the equation for the plate's coordinate

x_1

and solve it as a static constraint relating

x_1

x_2

. ((b)) Eliminate

x_1

and find the single surviving angular frequency of vertical oscillation.