I have finally gotten Carmen, my 3.2 kg pet monkey, off to sleep in her monkey-bouncer. The monkey bouncer is a 1.1 kg basket attached to the end of a vertical spring that has a k of 835 kg/s2. I know that my friend Puck is planning to sneak up and quickly pull down 15 cm on Carmen’s basket and let it go, so I want to install a shock absorber that will damp out Puck’s disturbance as quickly as possible. What value of b should I use for my shock absorber? How long will it take for Carmen’s displacement to decrease by a factor of 1/e2?
I have finally gotten Carmen, my 3.2 kg pet monkey, off to sleep in her monkey-bouncer. The monkey bouncer is a 1.1 kg basket attached to the end of a vertical spring that has a k of 835 kg/s2. I know that my friend Puck is planning to sneak up and quickly pull down 15 cm on Carmen’s basket and let it go, so I want to install a shock absorber that will damp out Puck’s disturbance as quickly as possible. What value of b should I use for my shock absorber? How long will it take for Carmen’s displacement to decrease by a factor of 1/e2?
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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I have finally gotten Carmen, my 3.2 kg pet monkey, off to sleep in her monkey-bouncer.
The monkey bouncer is a 1.1 kg basket attached to the end of a vertical spring that has a k of 835 kg/s2. I know that my friend Puck is planning to sneak up and quickly pull down 15 cm on Carmen’s basket and let it go, so I want to install a shock absorber that will damp out Puck’s disturbance as quickly as possible. What value of b should I use for my shock absorber? How long will it take for Carmen’s displacement to decrease by a factor of 1/e2?
![**Title: Damping Oscillations in a Monkey-Bouncer System**
**Problem Statement:**
A 3.2 kg pet monkey named Carmen is sleeping in a monkey-bouncer. The bouncer consists of a 1.1 kg basket attached to a vertical spring with a spring constant (k) of 835 kg/s². A friend, Puck, plans to pull down Carmen’s basket by 15 cm and release it, creating an oscillation. To minimize this disturbance, a shock absorber will be installed to damp the oscillations. Determine the damping constant (b) for the shock absorber. Additionally, find out how long it will take for Carmen’s displacement to decrease to 1/e of its initial value.
**Tasks:**
1. Sketch the situation and define all variables. Include a graph of the oscillation.
2. Identify the applicable physics equations and justify their use.
3. Apply these equations to solve for an algebraic expression and numerical answers. Clearly label and provide units for each answer.
**Analysis:**
1. **Variables:**
- Mass of monkey (m₁) = 3.2 kg
- Mass of basket (m₂) = 1.1 kg
- Total mass (m) = m₁ + m₂
- Spring constant (k) = 835 kg/s²
- Initial displacement (x₀) = 15 cm = 0.15 m
2. **Applicable Physics Equations:**
- Equation of motion for a damped harmonic oscillator:
\[
m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0
\]
- Damping ratio (ζ) and angular frequency (ω₀):
\[
ω₀ = \sqrt{\frac{k}{m}}
\]
\[
ζ = \frac{b}{2\sqrt{mk}}
\]
- Displacement as a function of time for underdamped motion:
\[
x(t) = x₀ e^{-ζω₀t} \cos(ω₁t + φ)
\]
where \(ω₁ = ω₀\sqrt{1-ζ^2}\)
**Calculations:**
3. **Solving:**
- Calculate total mass (m):
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Transcribed Image Text:**Title: Damping Oscillations in a Monkey-Bouncer System**
**Problem Statement:**
A 3.2 kg pet monkey named Carmen is sleeping in a monkey-bouncer. The bouncer consists of a 1.1 kg basket attached to a vertical spring with a spring constant (k) of 835 kg/s². A friend, Puck, plans to pull down Carmen’s basket by 15 cm and release it, creating an oscillation. To minimize this disturbance, a shock absorber will be installed to damp the oscillations. Determine the damping constant (b) for the shock absorber. Additionally, find out how long it will take for Carmen’s displacement to decrease to 1/e of its initial value.
**Tasks:**
1. Sketch the situation and define all variables. Include a graph of the oscillation.
2. Identify the applicable physics equations and justify their use.
3. Apply these equations to solve for an algebraic expression and numerical answers. Clearly label and provide units for each answer.
**Analysis:**
1. **Variables:**
- Mass of monkey (m₁) = 3.2 kg
- Mass of basket (m₂) = 1.1 kg
- Total mass (m) = m₁ + m₂
- Spring constant (k) = 835 kg/s²
- Initial displacement (x₀) = 15 cm = 0.15 m
2. **Applicable Physics Equations:**
- Equation of motion for a damped harmonic oscillator:
\[
m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0
\]
- Damping ratio (ζ) and angular frequency (ω₀):
\[
ω₀ = \sqrt{\frac{k}{m}}
\]
\[
ζ = \frac{b}{2\sqrt{mk}}
\]
- Displacement as a function of time for underdamped motion:
\[
x(t) = x₀ e^{-ζω₀t} \cos(ω₁t + φ)
\]
where \(ω₁ = ω₀\sqrt{1-ζ^2}\)
**Calculations:**
3. **Solving:**
- Calculate total mass (m):
\
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