Four spring/mass systems are shown in the figure. The systems have different masses, different spring constants, and are stretched different distances Ax, all as shown in the figure. Rank the systems, from smallest to largest, by their angular frequency. 5k 5 cm 4 cm 3 ст 2 cm Ax = 2m .5m 2m m C, D, B, A А, С, В, D D, C, B, A D, B, C, A А, В, С, D lll
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 Systems Analysis
Four spring/mass systems are shown in the figure. The systems have different masses, different spring constants, and are stretched different distances Δx, all as shown in the figure. Rank the systems, from smallest to largest, by their angular frequency.
#### Diagram Description
- **System A:**
- Spring constant: 5k
- Mass: m
- Stretched distance (Δx): 5 cm
- **System B:**
- Spring constant: 3k
- Mass: 2m
- Stretched distance (Δx): 4 cm
- **System C:**
- Spring constant: 2k
- Mass: 0.5m
- Stretched distance (Δx): 3 cm
- **System D:**
- Spring constant: k
- Mass: 2m
- Stretched distance (Δx): 2 cm
The image also includes a set of multiple-choice options asking you to rank the systems based on their angular frequency, defined as:
\[ \omega = \sqrt{\frac{k}{m}} \]
#### Choices for Ranking Angular Frequency:
- C, D, B, A
- A, C, B, D
- D, C, B, A
- D, B, C, A
- A, B, C, D
The correct choice can be determined by calculating the angular frequency for each system and comparing them:
- **System A:**
\[ \omega_A = \sqrt{\frac{5k}{m}} \]
- **System B:**
\[ \omega_B = \sqrt{\frac{3k}{2m}} \]
- **System C:**
\[ \omega_C = \sqrt{\frac{2k}{0.5m}} = \sqrt{\frac{4k}{m}} \]
- **System D:**
\[ \omega_D = \sqrt{\frac{k}{2m}} \]
### Step-by-Step Solution:
1. For **System A**:
\[
\omega_A = \sqrt{\frac{5k}{m}}
\]
2. For **System B**:
\[
\omega_B = \sqrt{\frac{3k}{2m}} = \sqrt{\frac{1.5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F21660ac5-28b2-4571-9555-dad5f0a9c878%2F65d99953-5d81-41e7-8cfe-73f440abb9f4%2Ftqc1naq_processed.png&w=3840&q=75)
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