N A mass of 1 kg is attached to the end of a spring whose restoring force is 180 The mass is in a medium m m . that exerts a viscous resistance of 156 N when the mass has a velocity of 6 — The viscous resistance is proportional to the speed of the object. S Suppose the spring is stretched 0.07 m beyond the its natural position and released. Let positive displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a force of 7 sin(3t) N at time t seconds. Find an function to express the steady-state component of the object's displacement from the spring's natural position, in m after t seconds. (Note: This spring-mass system is not "hanging", so there is no gravitational force included in the model.) u(t) = -13t(q + cos( √II t) + c₂ sin(√✓/II t))+ + (7 sin(3t) – 78 cos (3t)) x syntax error. 7 35325 - Check your variables - you might be using an incorrect one.
N A mass of 1 kg is attached to the end of a spring whose restoring force is 180 The mass is in a medium m m . that exerts a viscous resistance of 156 N when the mass has a velocity of 6 — The viscous resistance is proportional to the speed of the object. S Suppose the spring is stretched 0.07 m beyond the its natural position and released. Let positive displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a force of 7 sin(3t) N at time t seconds. Find an function to express the steady-state component of the object's displacement from the spring's natural position, in m after t seconds. (Note: This spring-mass system is not "hanging", so there is no gravitational force included in the model.) u(t) = -13t(q + cos( √II t) + c₂ sin(√✓/II t))+ + (7 sin(3t) – 78 cos (3t)) x syntax error. 7 35325 - Check your variables - you might be using an incorrect one.
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A mass of 1 kgkg is attached to the end of a spring whose restoring force is 180 NmNm. The mass is in a medium that exerts a viscous resistance of 156 NN when the mass has a velocity of 6 msms. The viscous resistance is proportional to the speed of the object.
Suppose the spring is stretched 0.07 mm beyond the its natural position and released. Let positive displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a force of 7sin(3t)7sin(3t) NN at time tt seconds.
Find an function to express the steady-state component of the object's displacement from the spring's natural position, in mm after tt seconds. (Note: This spring-mass system is not "hanging", so there is no gravitational force included in the model.)
u(t) = e−13t(c1+cos(√11t)+c2sin(√11t))+735325(7sin(3t)−78cos(3t))Incorrect syntax error. Check your variables - you might be using an incorrect one.
![### Spring-Mass System in a Viscous Medium
#### Problem Description:
A mass of \(1 \, \text{kg}\) is attached to the end of a spring with a restoring force of \(180 \, \frac{N}{m}\). The mass is in a medium that exerts a viscous resistance of \(156 \, \text{N}\) when the mass has a velocity of \(6 \, \frac{m}{s}\). The viscous resistance is directly proportional to the speed of the object.
#### Initial Conditions:
Suppose the spring is stretched \(0.07 \, \text{m}\) beyond its natural position and released. Here, positive displacements indicate a stretched spring. Additionally, external vibrations act on the mass with a force of \(7 \sin(3t) \, \text{N}\) at time \(t\) seconds.
#### Task:
Find a function to express the steady-state component of the object's displacement from the spring’s natural position, \(u(t)\), in meters at time \(t\) seconds. Note that this spring-mass system is not "hanging," so gravitational force is not included in the model.
#### Given Solution Formula:
\[ u(t) = e^{-13t} \left( c_1 + \cos\left(\sqrt{11} \, t \right) + c_2 \sin\left(\sqrt{11} \, t \right) \right) + \frac{7}{35325} \left( 7 \sin(3t) - 78 \cos(3t) \right) \]
**Error Notice:** The provided formula has a **syntax error** indicating possible incorrect variable usage.
#### Instructions:
**Check your variables** – you might be using an incorrect one. Specifically, ensure that constants \(c_1\) and \(c_2\) are appropriately determined by the initial conditions and the harmonic terms (sine and cosine functions) do not have syntax errors.
### Detailed Analysis
A graph or diagram is not provided in the text, but typically, you might find a graph illustrating the displacement \(u(t)\) over time \(t\). The graph would show oscillatory behavior dampened over time due to the exponential \(e^{-13t}\) term. Here's an explanation of each component in the provided formula:
- \( e^{-13t} \): Represents](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22e84192-4b16-4dc2-817c-afb771201e2b%2F99427274-8ee6-45a7-b61d-d07cf7ae2f49%2Fziqoo8w_processed.png&w=3840&q=75)
Transcribed Image Text:### Spring-Mass System in a Viscous Medium
#### Problem Description:
A mass of \(1 \, \text{kg}\) is attached to the end of a spring with a restoring force of \(180 \, \frac{N}{m}\). The mass is in a medium that exerts a viscous resistance of \(156 \, \text{N}\) when the mass has a velocity of \(6 \, \frac{m}{s}\). The viscous resistance is directly proportional to the speed of the object.
#### Initial Conditions:
Suppose the spring is stretched \(0.07 \, \text{m}\) beyond its natural position and released. Here, positive displacements indicate a stretched spring. Additionally, external vibrations act on the mass with a force of \(7 \sin(3t) \, \text{N}\) at time \(t\) seconds.
#### Task:
Find a function to express the steady-state component of the object's displacement from the spring’s natural position, \(u(t)\), in meters at time \(t\) seconds. Note that this spring-mass system is not "hanging," so gravitational force is not included in the model.
#### Given Solution Formula:
\[ u(t) = e^{-13t} \left( c_1 + \cos\left(\sqrt{11} \, t \right) + c_2 \sin\left(\sqrt{11} \, t \right) \right) + \frac{7}{35325} \left( 7 \sin(3t) - 78 \cos(3t) \right) \]
**Error Notice:** The provided formula has a **syntax error** indicating possible incorrect variable usage.
#### Instructions:
**Check your variables** – you might be using an incorrect one. Specifically, ensure that constants \(c_1\) and \(c_2\) are appropriately determined by the initial conditions and the harmonic terms (sine and cosine functions) do not have syntax errors.
### Detailed Analysis
A graph or diagram is not provided in the text, but typically, you might find a graph illustrating the displacement \(u(t)\) over time \(t\). The graph would show oscillatory behavior dampened over time due to the exponential \(e^{-13t}\) term. Here's an explanation of each component in the provided formula:
- \( e^{-13t} \): Represents
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