A 4-pound weight is hung on a spring and stretches it 1 foot. The mass spring system is then put into motion in a medium offering a damping force numerically equal to the velocity. If the mass is pulled down 6 inches from equilibrium and released, the initial value problem describing the position, $x ( t )$ , of the mass at time t is
A) $x ^ { \prime \prime } - 8 x ^ { \prime } + 32 x = 0 , x ( 0 ) = 6 , x ^ { \prime } ( 0 ) = 0$
B) $x ^ { \prime \prime } + 8 x ^ { \prime } + 32 x = 0 , x ( 0 ) = 6 , x ^ { \prime } ( 0 ) = 0$
C) $x ^ { \prime \prime } - 8 x ^ { \prime } + 32 x = 0 , x ( 0 ) = 1 / 2 , x ^ { \prime } ( 0 ) = 0$
D) $x ^ { \prime \prime } + 8 x ^ { \prime } + 32 x = 0 , x ( 0 ) = 1 / 2 , x ^ { \prime } ( 0 ) = 0$
E) $x ^ { \prime \prime } + 32 x = 8 , x ( 0 ) = 1 / 2 , x ^ { \prime } ( 0 ) = 0$

Question

Quizplus · Accepted Answer

The correct equation for a damped oscillator is $x'' + bx' + kx = 0$, where $b$ is the damping coefficient and $k$ is the spring constant. Given the damping force is numerically equal to the velocity, and the spring stretches 1 foot (or 12 inches) under a 4-pound weight, we can use $k = \frac{weight}{displacement} = \frac{4}{1} = 4$ pounds per foot. However, since we need $k$ in terms of pounds per inch for consistency with the initial displacement given in inches, we convert it to $k = 4 \times 12 = 48$ pounds per inch. The mass $m = \frac{4}{32.2}$ slugs (using $g \approx 32.2 \text{ft/s}^2$). The equation of motion becomes $mx'' + bx' + kx = 0$. To find the equivalent $k$ and $b$ in terms of the system given, we normalize $k$ and $b$ by $m$, leading to a system that matches the form given in the options. The initial condition is given as 6 inches from equilibrium, which is $1/2$ foot, so $x(0) = 1/2$, and since it's released from rest, $x'(0) = 0$. The correct equation must have positive coefficients for $x'$ and $x$ due to the nature of damping and spring forces, and the initial conditions must match the physical scenario described.

A 4-Pound Weight Is Hung on a Spring and Stretches x(t)x ( t )x(t)

A 4-Pound Weight Is Hung on a Spring and Stretches $x ( t )$