A spring with a mass of $2 \mathrm {~kg}$ has damping constant 8 and spring constant 80 . Graph the position function of the mass at time $t$ if it starts at the equilibrium position with a velocity of $2 \mathrm {~m} / \mathrm { s }$.
Select the correct answer.

Question

Quizplus · Accepted Answer

The position function of the mass can be given by:

$$x(t) = e^{-\frac{8}{2}t} \left(A \cos(3t) + B \sin(3t)\right)$$

where $A$ and $B$ are constants determined by the initial conditions.

Since the mass starts at equilibrium with a velocity of 2 m/s, we have:

$$x(0) = 0, \quad x'(0) = 2$$

Substituting these initial conditions into the position function, we get:

$$0 = A, \quad 2 = 3B - 4A$$

Solving for $A$ and $B$, we obtain:

$$A = 0, \quad B = \frac{2}{3}$$

Thus, the position function of the mass is:

$$x(t) = e^{-4t}\left(\frac{2}{3}\sin(3t)\right)$$

We can graph this function using a computer or calculator, and the resulting graph matches choice B.

A Spring with a Mass Of 2 kg2 \mathrm {~kg}2 kg Has Damping Constant 8 and Spring Constant 80

A Spring with a Mass Of $2 \mathrm {~kg}$ Has Damping Constant 8 and Spring Constant 80