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. 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 8² gravity. u(t) =

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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(
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
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
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
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