Question

Problem 3 (35 points) (a) Derive the equation of motion for the frictionless spring-mass ($k$, $m$) system shown in the figure in terms of $x$, displacement of the mass measured from the static equilibrium position. \text{ -- using a free body diagram} \text{ -- using the principle of conservation of energy} The angle of the wedge is $\alpha$. Acceleration of gravity is $g$ (b) Determine the natural frequency of vibration. (c) In reality, the mass slides on the wedge with a coefficient of friction $\mu$ (Coulomb friction). The mass is now displaced 25 $mm$ from the static equilibrium condition and released. It is observed that the amplitude decreases 1.2 $mm$ each cycle. What is the coefficient of friction $\mu$? $m = 10\text{ kg}$, $k = 10,000\text{ N/m}$, $\alpha = 30^\circ$, $g = 9.8\text{ m/s}^2$

          Problem 3 (35 points)
(a) Derive the equation of motion for the frictionless spring-mass ($k$, $m$) system shown in the
figure in terms of $x$, displacement of the mass measured from the static equilibrium position.
\text{ -- using a free body diagram}
\text{ -- using the principle of conservation of energy}
The angle of the wedge is $\alpha$. Acceleration of gravity is $g$
(b) Determine the natural frequency of vibration.
(c) In reality, the mass slides on the wedge with a coefficient of friction $\mu$ (Coulomb friction).
The mass is now displaced 25 $mm$ from the static equilibrium condition and released. It
is observed that the amplitude decreases 1.2 $mm$ each cycle. What is the coefficient of
friction $\mu$?
$m = 10\text{ kg}$, $k = 10,000\text{ N/m}$, $\alpha = 30^\circ$, $g = 9.8\text{ m/s}^2$

Problem 3 (35 points)
(a) Derive the equation of motion for the frictionless spring-mass (k, m) system shown in the
figure in terms of x, displacement of the mass measured from the static equilibrium position.
– using a free body diagram
– using the principle of conservation of energy
The angle of the wedge is α. Acceleration of gravity is g
(b) Determine the natural frequency of vibration.
(c) In reality, the mass slides on the wedge with a coefficient of friction μ (Coulomb friction).
The mass is now displaced 25 mm from the static equilibrium condition and released. It
is observed that the amplitude decreases 1.2 mm each cycle. What is the coefficient of
friction μ?
m = 10 kg, k = 10,000 N/m, α = 30^∘, g = 9.8 m/s^2

Added by Alex M.

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