8. A spring (k = 40 N/m) is mounted to a table. The unelongated length of the spring is 40 cm. A 0.026 kg block is gently placed on top of the spring. After some time passes, the block will be at a certain minimum distance above the table. Determine the compression of the spring at that time. A. 0.026 m B. 0.013 m C. 0.006 m. D. 0.003 m
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 Description
A spring with a spring constant \( k = 40 \, \text{N/m} \) is mounted on a table. The unelongated length of the spring is 40 cm. A block weighing 0.026 kg is gently placed on top of the spring. After some time, the block will stabilize at a certain minimum distance above the table. Determine the compression of the spring at that time.
### Possible Answers
A. \( 0.026 \, \text{m} \)
B. \( 0.013 \, \text{m} \)
C. \( 0.006 \, \text{m} \)
D. \( 0.003 \, \text{m} \)
### Explanation
To solve this problem, apply Hooke's Law and balance the forces acting on the block. The gravitational force on the block equals the spring force at equilibrium.
1. **Gravitational Force**: \( F_g = m \cdot g \)
Where:
- \( m = 0.026 \, \text{kg} \)
- \( g = 9.8 \, \text{m/s}^2 \)
2. **Spring Force**: \( F_s = k \cdot x \)
Where:
- \( k = 40 \, \text{N/m} \)
- \( x \) is the compression of the spring.
Set \( F_g = F_s \) and solve for \( x \).
\[
m \cdot g = k \cdot x \\
0.026 \cdot 9.8 = 40 \cdot x \\
x = \frac{0.26}{40} \\
x = 0.0065 \, \text{m}
\]
Thus, the correct answer is close to option C, \( 0.006 \, \text{m} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf9dafc6-bac8-41ec-a856-62d58445bc24%2F1de031ad-ffd0-4a6d-8ba7-ac9d1e68d241%2Ftxl3loe_processed.jpeg&w=3840&q=75)
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