A horizontal spring is lying on a frictionless surface. One end of the spring is attaches to a wall while the other end is connected to a movable object. The spring and object are compressed by 0.058 m, released from rest, and subsequently oscillate back and forth with an angular frequency of 11.0 rad/s. What is the speed of the object at the instant when the spring is stretched by 0.031 m relative to its unstrained length?
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
![**Problem Statement:**
A horizontal spring is lying on a frictionless surface. One end of the spring is attached to a wall while the other end is connected to a movable object. The spring and object are compressed by \(0.058 \, \text{m}\), released from rest, and subsequently oscillate back and forth with an angular frequency of \(11.0 \, \text{rad/s}\). What is the speed of the object at the instant when the spring is stretched by \(0.031 \, \text{m}\) relative to its unstrained length?
**Solution:**
To solve this problem, we will use the principles of simple harmonic motion and the conservation of mechanical energy.
1. **Given Data:**
- Initial compression of the spring, \( x_{\text{initial}} = 0.058 \, \text{m} \)
- Frequency of oscillation, \( \omega = 11.0 \, \text{rad/s} \)
- Stretch distance at the moment of interest, \( x = 0.031 \, \text{m} \)
2. **Conservation of Energy:**
The total mechanical energy in a simple harmonic oscillator consists of potential energy in the spring and the kinetic energy of the mass.
- **Total Mechanical Energy (E):**
\[
E = \frac{1}{2} k x_{\text{initial}}^2
\]
where \( k \) is the spring constant.
- **Potential Energy at x (U):**
\[
U = \frac{1}{2} k x^2
\]
3. **Spring Constant (k):**
Since \(\omega = \sqrt{\frac{k}{m}}\), we can express \( k \) in terms of \(\omega\) and \( m \):
\[
k = m \omega^2
\]
4. **Kinetic Energy (K):**
The difference between the total mechanical energy and the potential energy gives us the kinetic energy:
\[
K = E - U = \frac{1}{2} k x_{\text{initial}}^2 - \frac{1}{2} k x^2
\]
Simplifying:
\[
K = \frac{1}{2} k \left( x_{\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda579a50-4d79-460d-b1ac-a80a17754259%2Fa2b6d178-f905-4485-a058-58ec7e4a2919%2Ffxpddad_processed.png&w=3840&q=75)
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