Suppose a 32-pound weight is suspended on a spring.The weight is pulled foot below the equilibrium position and released.The motion takes place in a Med that furnishes a
Damping force of magnitude speed at all times.Assume that the weight stretches the spring foot
From its natural position.Find a formula for the position of the weight as a function of time t.

Question

Suppose a 32-pound weight is suspended on a spring.The weight is pulled  foot below the equilibrium position and released.The motion takes place in a Med that furnishes a
Damping force of magnitude  speed at all times.Assume that the weight stretches the spring  foot
From its natural position.Find a formula for the position of the weight as a function of time t.

Quizplus · Accepted Answer

The position of the weight as a function of time t can be represented by the following equation:
x(t) = A cos(ωt) + B sin(ωt) + C
where 
A = amplitude of the motion = 11ebbd4d_3f18_e74b_8a0a_070b9336c882_TB8202_11
ω = angular frequency = √(k/m) = √(g/l) = √(32/11ebbd4d_3f18_e74d_8a0a_ffa017fbc963_TB8202_11) = 11ebbd4d_3f19_0e63_8a0a_565d452ab0bc_TB8202_11
B = 0 (since there is no initial velocity)
C = equilibrium position = 11ebbd4d_3f19_0e60_8a0a_d7b4f8a119ca_TB8202_11

To find the value of A, observe that the weight is pulled 11ebbd4d_3f18_e74b_8a0a_070b9336c882_TB8202_11 foot below the equilibrium position and released. Thus, the initial displacement is -11ebbd4d_3f18_e74b_8a0a_070b9336c882_TB8202_11, which gives:
x(0) = A cos(0) + B sin(0) + C = -11ebbd4d_3f18_e74b_8a0a_070b9336c882_TB8202_11
Simplifying, we get:
A = -11ebbd4d_3f19_0e64_8a0a_7115788e4734_TB8202_11

Therefore, the equation for the position of the weight as a function of time t is:
x(t) = -11ebbd4d_3f19_0e64_8a0a_7115788e4734_TB8202_11 cos(11ebbd4d_3f19_0e63_8a0a_565d452ab0bc_TB8202_11 t) + 11ebbd4d_3f19_0e60_8a0a_d7b4f8a119ca_TB8202_11

Simplifying the expression, we get:
x(t) = 11ebbd4d_3f18_e74c_8a0a_3d0014d4a906_TB8202_11 cos(11ebbd4d_3f19_0e63_8a0a_565d452ab0bc_TB8202_11 t) + 11ebbd4d_3f19_0e60_8a0a_d7b4f8a119ca_TB8202_11

Thus, the correct choice is C. The motion of the weight can be described by the equation $y(t) = Ae^{-bt}\sin(\omega t+\phi)$, where $A$ is the amplitude, $b$ is the damping constant, $\omega$ is the angular frequency, and $\phi$ is the phase angle. We can use the given information to determine these values:

- The amplitude is 1/2 of the distance the weight is pulled below the equilibrium position, or $A=\frac{1}{2}(11ebbd4d_3f18_e74b_8a0a_070b9336c882_TB8202_11)=11ebbd4d_3f18_e74c_8a0a_57d879a65e8c_TB8202_11$.
- The damping constant is equal to the damping force divided by the mass, or $b=\frac{11ebbd4d_3f18_e74c_8a0a_57d879a65e8c_TB8202_11}{32}$.
- The angular frequency is $\omega=\sqrt{\frac{k}{m}-\left(\frac{b}{2m}ight)^2}$, where $k$ is the spring constant and $m$ is the mass. We can use the given information to find $k$: $mg=k(11ebbd4d_3f18_e74d_8a0a_ffa017fbc963_TB8202_11)$, so $k=\frac{mg}{11ebbd4d_3f18_e74d_8a0a_ffa017fbc963_TB8202_11}$. Substituting this into the formula for $\omega$ gives $\omega=\sqrt{\frac{mg}{32}\left(\frac{1}{11ebbd4d_3f18_e74d_8a0a_ffa017fbc963_TB8202_11}ight)-\left(\frac{\frac{11ebbd4d_3f18_e74c_8a0a_57d879a65e8c_TB8202_11}{32}}{2	imes 32}ight)^2}$.
- The phase angle is $\phi=	an^{-1}\left(\frac{b\omega}{k}ight)$.

Putting it all together, we have:

$$y(t)=11ebbd4d_3f18_e74c_8a0a_57d879a65e8c_TB8202_11 e^{-\frac{11ebbd4d_3f18_e74c_8a0a_57d879a65e8c_TB8202_11}{32}t}\sin\left(\sqrt{\frac{mg}{32}\left(\frac{1}{11ebbd4d_3f18_e74d_8a0a_ffa017fbc963_TB8202_11}ight)-\left(\frac{\frac{11ebbd4d_3f18_e74c_8a0a_57d879a65e8c_TB8202_11}{32}}{2	imes 32}ight)^2}t+	an^{-1}\left(\frac{\frac{11ebbd4d_3f18_e74c_8a0a_57d879a65e8c_TB8202_11}{32}\sqrt{\frac{mg}{32}\left(\frac{1}{11ebbd4d_3f18_e74d_8a0a_ffa017fbc963_TB8202_11}ight)-\left(\frac{\frac{11ebbd4d_3f18_e74c_8a0a_57d879a65e8c_TB8202_11}{32}}{2	imes 32}ight)^2}}{\frac{mg}{11ebbd4d_3f18_e74d_8a0a_ffa017fbc963_TB8202_11}}ight)$$

Simplifying this expression is not necessary, and the exact numerical values are not needed for the answer choice. We can see that the correct answer is (C) by recognizing that it is the only choice that includes all the given information and describes the motion of a weight on a spring with damping.

Suppose a 32-Pound Weight Is Suspended on a Spring