Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Understanding Harmonic Motion with Damping
A mass of 0.75 kg stretches a spring 0.05 m. The mass is in a medium that exerts a viscous resistance of 57 N when the mass has a velocity of \( 6 \frac{m}{s} \). The viscous resistance is proportional to the speed of the object.
Suppose the object is displaced an additional 0.07 m and released.
Find a function to express the object's displacement from the spring's natural position, in meters after \( t \) seconds. Let positive displacements indicate a stretched spring, and use \( 9.8 \frac{m}{s^2} \) as the acceleration due to gravity.
\[ u(t) = \boxed{} \]
The given problem involves the following important details:
1. **Mass:** 0.75 kg.
2. **Initial Stretch of Spring:** 0.05 m.
3. **Viscous Resistance:** 57 N at a velocity of \( 6 \frac{m}{s} \).
4. **Additional Displacement:** 0.07 m.
5. **Gravitational Acceleration:** \( 9.8 \frac{m}{s^2} \).
To find the function \( u(t) \), we need to consider the system's damping and harmonic motion properties, which typically lead to a differential equation describing the mass-spring system.
### Steps to Solve:
1. **Determine Spring Constant (k):**
- \( mg = kx \Rightarrow k = \frac{mg}{x} = \frac{0.75 \times 9.8}{0.05} = 147 \, N/m \).
2. **Determine Damping Coefficient (c):**
- \( c = \frac{57 N}{6 \frac{m}{s}} = 9.5 \frac{Ns}{m} \).
3. **Formulate the Differential Equation:**
- The equation of motion for a damped harmonic oscillator:
\[ m\ddot{u} + c\dot{u} + ku = 0 \].
- Plug in the values:
\[ 0.75\ddot{u} + 9.5\dot{u} + 147u = 0 \].
4. **Initial Conditions:**
- \( u(](https://content.bartleby.com/qna-images/question/a86b2d75-dc89-44ee-816d-e64b811f53cb/b46966f7-ac5a-45d9-a12d-92726d208228/gipw3ck_processed.png)
Transcribed Image Text:### Understanding Harmonic Motion with Damping
A mass of 0.75 kg stretches a spring 0.05 m. The mass is in a medium that exerts a viscous resistance of 57 N when the mass has a velocity of \( 6 \frac{m}{s} \). The viscous resistance is proportional to the speed of the object.
Suppose the object is displaced an additional 0.07 m and released.
Find a function to express the object's displacement from the spring's natural position, in meters after \( t \) seconds. Let positive displacements indicate a stretched spring, and use \( 9.8 \frac{m}{s^2} \) as the acceleration due to gravity.
\[ u(t) = \boxed{} \]
The given problem involves the following important details:
1. **Mass:** 0.75 kg.
2. **Initial Stretch of Spring:** 0.05 m.
3. **Viscous Resistance:** 57 N at a velocity of \( 6 \frac{m}{s} \).
4. **Additional Displacement:** 0.07 m.
5. **Gravitational Acceleration:** \( 9.8 \frac{m}{s^2} \).
To find the function \( u(t) \), we need to consider the system's damping and harmonic motion properties, which typically lead to a differential equation describing the mass-spring system.
### Steps to Solve:
1. **Determine Spring Constant (k):**
- \( mg = kx \Rightarrow k = \frac{mg}{x} = \frac{0.75 \times 9.8}{0.05} = 147 \, N/m \).
2. **Determine Damping Coefficient (c):**
- \( c = \frac{57 N}{6 \frac{m}{s}} = 9.5 \frac{Ns}{m} \).
3. **Formulate the Differential Equation:**
- The equation of motion for a damped harmonic oscillator:
\[ m\ddot{u} + c\dot{u} + ku = 0 \].
- Plug in the values:
\[ 0.75\ddot{u} + 9.5\dot{u} + 147u = 0 \].
4. **Initial Conditions:**
- \( u(
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Follow-up Question
Transcribed Image Text:A mass of 0.75 kg stretches a spring 0.05 m. The mass is in a medium that exerts a viscous resistance of 57
m
N when the mass has a velocity of 6 The viscous resistance is proportional to the speed of the object.
S
Suppose the object is displaced an additional 0.07 m and released.
m
Find an function to express the object's displacement from the spring's natural position, in m after t
seconds. Let positive displacements indicate a stretched spring, and use 9.8 as the acceleration due to
s²
gravity.
u(t) = .07(e-6.33t) sin(1
-6.33t) sin 12.487t +
K|2
X
Solution
by Bartleby Expert
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
Transcribed Image Text:A mass of 0.75 kg stretches a spring 0.05 m. The mass is in a medium that exerts a viscous resistance of 57
m
N when the mass has a velocity of 6 The viscous resistance is proportional to the speed of the object.
S
Suppose the object is displaced an additional 0.07 m and released.
m
Find an function to express the object's displacement from the spring's natural position, in m after t
seconds. Let positive displacements indicate a stretched spring, and use 9.8 as the acceleration due to
s²
gravity.
u(t) = .07(e-6.33t) sin(1
-6.33t) sin 12.487t +
K|2
X
Solution
by Bartleby Expert
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