(a) You displace the object from x = 0 to ₁ > 0, holding the object steady at the beginning and end. What is the work done by the spring? What is the work done by you? (b) You let go of the spring and observe the object move from location ₁ to 2. Is 2 less than or greater than ₁? (e) What is the kinetic energy of the object at 2₂? (d) Is this bizarre spring realistic? Explain.
(a) You displace the object from x = 0 to ₁ > 0, holding the object steady at the beginning and end. What is the work done by the spring? What is the work done by you? (b) You let go of the spring and observe the object move from location ₁ to 2. Is 2 less than or greater than ₁? (e) What is the kinetic energy of the object at 2₂? (d) Is this bizarre spring realistic? Explain.
College Physics
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ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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Please solve this question using only algebra, no calculus
![### Problem 3: Analysis of a Bizarre Spring System
Suppose you have an object attached to a bizarre spring, initially located at \( x = 0 \). The equilibrium point of the force of the spring on the object is given by
\[ F_s = kx, \]
where \( k \) is a spring constant.
#### Questions and Explanations:
**(a)** You displace the object from \( x = 0 \) to \( x_1 > 0 \), holding the object steady at the beginning and end. What is the work done by the spring? What is the work done by you?
**(b)** You let go of the spring and observe the object move from location \( x_1 \) to \( x_2 \). Is \( x_2 \) less than or greater than \( x_1 \)?
**(c)** What is the kinetic energy of the object at \( x_2 \)?
**(d)** Is this bizarre spring realistic? Explain.
### Detailed Explanations:
**(a)** When you displace the object from \( x = 0 \) to \( x_1 \), the spring exerts a force that opposes this displacement. The work done by the spring is the work done against this force:
\[ W_{\text{spring}} = - \frac{1}{2} k x_1^2 \]
The work done by you to move the object:
\[ W_{\text{you}} = \frac{1}{2} k x_1^2 \]
This accounts for the work needed to hold the object in the displaced position against the spring force.
**(b)** Upon letting go of the spring, the object will move due to the restoring force of the spring. The location \( x_2 \) will satisfy energy conservation.
Initially, all the energy is potential:
\[ E_i = \frac{1}{2} k x_1^2 \]
At \( x_2 \), the energy will be a mix of potential and kinetic energy. Given the spring is linear, \( x_2 \) could be less than or greater than \( x_1 \) when considering overshooting and oscillations.
**(c)** The kinetic energy at \( x_2 \) can be derived from energy conservation principles:
\[ K_x2 = \frac{1}{2} k x_1^2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc28413f4-ef64-4ffe-aa5b-c6ada5af5391%2F7bb95c6d-04fd-409c-8fbc-bf6597793e96%2Fuama4sq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 3: Analysis of a Bizarre Spring System
Suppose you have an object attached to a bizarre spring, initially located at \( x = 0 \). The equilibrium point of the force of the spring on the object is given by
\[ F_s = kx, \]
where \( k \) is a spring constant.
#### Questions and Explanations:
**(a)** You displace the object from \( x = 0 \) to \( x_1 > 0 \), holding the object steady at the beginning and end. What is the work done by the spring? What is the work done by you?
**(b)** You let go of the spring and observe the object move from location \( x_1 \) to \( x_2 \). Is \( x_2 \) less than or greater than \( x_1 \)?
**(c)** What is the kinetic energy of the object at \( x_2 \)?
**(d)** Is this bizarre spring realistic? Explain.
### Detailed Explanations:
**(a)** When you displace the object from \( x = 0 \) to \( x_1 \), the spring exerts a force that opposes this displacement. The work done by the spring is the work done against this force:
\[ W_{\text{spring}} = - \frac{1}{2} k x_1^2 \]
The work done by you to move the object:
\[ W_{\text{you}} = \frac{1}{2} k x_1^2 \]
This accounts for the work needed to hold the object in the displaced position against the spring force.
**(b)** Upon letting go of the spring, the object will move due to the restoring force of the spring. The location \( x_2 \) will satisfy energy conservation.
Initially, all the energy is potential:
\[ E_i = \frac{1}{2} k x_1^2 \]
At \( x_2 \), the energy will be a mix of potential and kinetic energy. Given the spring is linear, \( x_2 \) could be less than or greater than \( x_1 \) when considering overshooting and oscillations.
**(c)** The kinetic energy at \( x_2 \) can be derived from energy conservation principles:
\[ K_x2 = \frac{1}{2} k x_1^2
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