A spring is attached to the ceiling and pulled 10 cm down from equilibrium and released. The amplitude decreases by 13% each second. The spring oscillates 9 times each second. Find an equation for the distance, D the end of the spring is below equilibrium in terms of seconds, t. D(t) =
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.
![**Spring Oscillation Problem**
A spring is attached to the ceiling and pulled 10 cm down from equilibrium and released. The amplitude decreases by 13% each second. The spring oscillates 9 times each second. Find an equation for the distance, \( D \), the end of the spring is below equilibrium in terms of seconds, \( t \).
\[ D(t) = \]
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To solve this problem, we need to create an equation for the motion of the spring.
### Key Components:
- **Initial Amplitude:** 10 cm
- **Damping Factor:** The amplitude decreases by 13% each second, so it retains 87% (100% - 13%) of its amplitude every second.
- **Frequency:** The spring oscillates 9 times per second.
### Steps to Create the Equation
1. **Exponential Decay of Amplitude:** The amplitude decreases with a factor of 0.87 each second. Therefore, at any time \( t \), the amplitude will be \( 10 \times 0.87^t \).
2. **Oscillation Component:** Since the spring oscillates 9 times per second, the angular frequency \( \omega \) can be calculated as:
\[ \omega = 2\pi \times 9 \]
3. **Resulting Equation:** The distance \( D(t) \) below the equilibrium position is given by:
\[ D(t) = 10 \times 0.87^t \times \sin(2\pi \times 9t) \]
This equation takes into account both the decay of the amplitude due to damping and the frequency of oscillation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c16c5d2-e997-4e04-a749-57ee722e7387%2Fcad573f6-4a58-44f6-876d-6525ec8b88f6%2Fzs9ps4w_processed.png&w=3840&q=75)
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