A spring is attached to the ceiling and pulled 10 cm down from equilibrium and released. The amplitude decreases by 11% 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.
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.
![A spring is attached to the ceiling and pulled 10 cm down from equilibrium and released. The amplitude decreases by 11% 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, \( t \).
\[ D(t) = \]
Explanation:
This problem involves the motion of a damped harmonic oscillator. The spring's initial displacement, amplitude decay, and oscillation frequency create a scenario where you can model the motion mathematically.
1. **Initial Conditions**:
- Initial displacement: 10 cm
- Amplitude decay per second: 11%
- Frequency of oscillation: 9 cycles per second
2. **Equation Components**:
- **Amplitude Decay**: The exponential decay formula is used for reducing amplitude. If the amplitude decreases by 11% each second, the decay factor per second is \( 1 - 0.11 = 0.89 \).
- **Oscillation**: The oscillation can be described using a sine or cosine function. The angular frequency \( \omega \) is calculated from the frequency as \( \omega = 2\pi \times \text{frequency} \).
3. **General Form of the Equation**:
- The equation is in the form \( D(t) = A_0 \times (0.89)^t \times \cos(\omega t) \) or using sine, depending on the initial phase.
4. **Calculating Components**:
- Initial Amplitude (\( A_0 \)): 10 cm
- Amplitude reduction factor per second: 0.89
- Frequency: 9 Hz, so \( \omega = 2\pi \times 9 \)
5. **Final Equation**:
- \( D(t) = 10 \times (0.89)^t \times \cos(18\pi t) \)
The equation \( D(t) \) models the distance of the spring's end below equilibrium at any time \( t \), accounting for both the oscillatory nature and the damping effect from amplitude reduction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c16c5d2-e997-4e04-a749-57ee722e7387%2Fddadfeef-8ba5-43a7-b400-b690eec70180%2Fk0p2f8j_processed.png&w=3840&q=75)
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