Question

3. A spring is 4 cm long in its unstretched position, but stretches to an equilibrium length of 8 cm when a mass of $\frac{1}{3}$ kg is hung from it. We'll use the following notation: • $L = 4$ cm, the unstretched length of the spring. • $s = 4$ cm, the total amount of stretching from the unstretched position to the equilibrium position. • $x(t) = $ any amount of additional stretching in the spring at time $t$. (So positive $x(t)$ means that the spring is longer than 8 cm at time $t$ and the mass is below the equilibrium position, while negative $x(t)$ means that the spring is shorter than 8 cm at time $t$ and the mass is above the equilibrium position.) Complete the following tasks: (a) Find the spring constant $k$, using Hooke's law (and the fact that the force of gravity balances the restorative force when the weight is hung from the spring). Your answer should be in N/m, where $N = \frac{kg}{sec^2}$. You may approximate the acceleration $g$ due to gravity by $g = 9.8 m/sec^2$, or you can write your answer in terms of $g$. (b) Assuming the spring obeys Hooke's law, the restorative force of the spring, which opposes any stretching or compression away from the unstretched configuration, is $F_{rest}(t) = -k(x(t) + s)$ at time $t$. Explain why (neglecting damping) the net amount of force on the mass at time $t$ is $-kx(t)$ (rather than $-k(x(t) + s)$). (c) Assume the spring is given an initial velocity of 5 cm/second downward, starting from the equilibrium position $x(0) = 0$, and that it experiences a damping force of $F_{damping} = -\sqrt{2}v$, where $v$ is the velocity of the spring. Write down an initial value problem that describes the setup. (d) Does this setup correspond to underdamping, overdamping, or critical damping? Briefly justify your answer. (e) Solve the IVP in part (c). Note: Some of the constants are a bit messy. For this problem, you may use a calculator to approximate them if you wish. (Side note: You should always use exact answers in this course except where explicitly instructed otherwise, as is the case here.)

          3. A spring is 4 cm long in its unstretched position, but stretches to an equilibrium length of 8 cm when a mass of $\frac{1}{3}$ kg is hung from it. We'll use the following notation:
• $L = 4$ cm, the unstretched length of the spring.
• $s = 4$ cm, the total amount of stretching from the unstretched position to the equilibrium position.
• $x(t) = $ any amount of additional stretching in the spring at time $t$. (So positive $x(t)$ means that the spring is longer than 8 cm at time $t$ and the mass is below the equilibrium position, while negative $x(t)$ means that the spring is shorter than 8 cm at time $t$ and the mass is above the equilibrium position.)
Complete the following tasks:
(a) Find the spring constant $k$, using Hooke's law (and the fact that the force of gravity balances the restorative force when the weight is hung from the spring). Your answer should be in N/m, where $N = \frac{kg}{sec^2}$. You may approximate the acceleration $g$ due to gravity by $g = 9.8 m/sec^2$, or you can write your answer in terms of $g$.
(b) Assuming the spring obeys Hooke's law, the restorative force of the spring, which opposes any stretching or compression away from the unstretched configuration, is $F_{rest}(t) = -k(x(t) + s)$ at time $t$. Explain why (neglecting damping) the net amount of force on the mass at time $t$ is $-kx(t)$ (rather than $-k(x(t) + s)$).
(c) Assume the spring is given an initial velocity of 5 cm/second downward, starting from the equilibrium position $x(0) = 0$, and that it experiences a damping force of $F_{damping} = -\sqrt{2}v$, where $v$ is the velocity of the spring. Write down an initial value problem that describes the setup.
(d) Does this setup correspond to underdamping, overdamping, or critical damping? Briefly justify your answer.
(e) Solve the IVP in part (c). Note: Some of the constants are a bit messy. For this problem, you may use a calculator to approximate them if you wish. (Side note: You should always use exact answers in this course except where explicitly instructed otherwise, as is the case here.)

3. A spring is 4 cm long in its unstretched position, but stretches to an equilibrium length of 8 cm when a mass of (1)/(3) kg is hung from it. We'll use the following notation:
• L = 4 cm, the unstretched length of the spring.
• s = 4 cm, the total amount of stretching from the unstretched position to the equilibrium position.
• x(t) = any amount of additional stretching in the spring at time t. (So positive x(t) means that the spring is longer than 8 cm at time t and the mass is below the equilibrium position, while negative x(t) means that the spring is shorter than 8 cm at time t and the mass is above the equilibrium position.)
Complete the following tasks:
(a) Find the spring constant k, using Hooke's law (and the fact that the force of gravity balances the restorative force when the weight is hung from the spring). Your answer should be in N/m, where N = (kg)/(sec^2). You may approximate the acceleration g due to gravity by g = 9.8 m/sec^2, or you can write your answer in terms of g.
(b) Assuming the spring obeys Hooke's law, the restorative force of the spring, which opposes any stretching or compression away from the unstretched configuration, is Frest(t) = -k(x(t) + s) at time t. Explain why (neglecting damping) the net amount of force on the mass at time t is -kx(t) (rather than -k(x(t) + s)).
(c) Assume the spring is given an initial velocity of 5 cm/second downward, starting from the equilibrium position x(0) = 0, and that it experiences a damping force of Fdamping = -√(2)v, where v is the velocity of the spring. Write down an initial value problem that describes the setup.
(d) Does this setup correspond to underdamping, overdamping, or critical damping? Briefly justify your answer.
(e) Solve the IVP in part (c). Note: Some of the constants are a bit messy. For this problem, you may use a calculator to approximate them if you wish. (Side note: You should always use exact answers in this course except where explicitly instructed otherwise, as is the case here.)

Added by Katherine R.

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