A mass weighting 64 lbs stretches a spring 6 inches. The mass is in a medium that exerts a viscous resistance of 9 lbs when the mass has a velocity of 6 ft/sec. Suppere the obiect is displaced an additional 7 inches and releaced
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-Mass-Damping System Analysis
**Problem Statement:**
A mass weighing 64 lbs stretches a spring 6 inches. The mass is in a medium that exerts a viscous resistance of 9 lbs when the mass has a velocity of 6 ft/sec.
Suppose the object is displaced an additional 7 inches and released.
**Goal:**
Find an equation for the object's displacement, \( u(t) \), in feet after \( t \) seconds.
**Equation to Determine**:
\[ u(t) = \ \text{[Enter equation here]} \]
**Given Data:**
1. **Mass of the Object:** 64 lbs
2. **Spring Stretch:** 6 inches (0.5 feet)
3. **Viscous Resistance:** 9 lbs at 6 ft/sec velocity
4. **Additional Displacement:** 7 inches (0.5833 feet)
**Instructions:**
To find the displacement equation \( u(t) \), we need to use principles from physics, particularly the concepts concerning harmonic motion in a damped system. This problem involves solving a differential equation, typically of the form:
\[ m \frac{d^2u}{dt^2} + c \frac{du}{dt} + ku = 0\]
where:
- \( m \) is the mass,
- \( c \) is the damping coefficient,
- \( k \) is the spring constant,
- \( u \) is the displacement from equilibrium.
By analyzing the described system, taking into account the given parameters and applying the appropriate damping and oscillation formulas, we will derive the corresponding \( u(t) \).
Feel free to fill in the derived equation in the provided place once the differential equations have been solved. For further understanding, refer to resources on harmonic oscillators, damped motion, and differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc700bc18-61e4-4e04-a31d-52f10b21c2d0%2F0fc6d87b-c80d-41b3-b60b-732e58a33fc0%2F9grm3v_processed.jpeg&w=3840&q=75)
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