water fills a leng th l of a U tube. The water is slightly displaced and then allowed to move freely. (A.) Show that the liquid executes Simple harmonic motion (B.) Whot is the periad ?
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.
![**Topic: Simple Harmonic Motion of Liquid in a U-Tube**
**Introduction:**
When liquid fills a length \( l \) of a U-tube, and this liquid is slightly displaced and then allowed to move freely, it exhibits simple harmonic motion (SHM). Understanding this phenomenon requires demonstrating the conditions under which the liquid executes SHM and determining the period of this oscillation.
**Part A: Show that the Liquid Executes Simple Harmonic Motion**
To demonstrate that the liquid in the U-tube executes simple harmonic motion, consider the following steps:
1. **Initial Displacement:**
When the liquid in the U-tube is displaced by a small distance \( x \):
\[ \text{Restoring force} (F) = -kx \]
where \( k \) is the spring constant, analogous to Hooke's law for a spring.
2. **Restoring Force and Acceleration:**
The restoring force is provided by the difference in hydrostatic pressure on either side of the U-tube. When the liquid is displaced by \( x \), the height difference between the two columns of liquid causes a pressure difference, which in turn causes the liquid to return to its equilibrium position.
3. **Deriving the Equations of Motion:**
Let's denote \( \rho \) as the density of the liquid, \( g \) as the acceleration due to gravity, and \( A \) as the cross-sectional area of the U-tube.
The restoring force acting on the liquid is proportional to the displacement \( x \):
\[ F = -2\rho g A x \]
The factor of 2 comes from the fact that the displacement \( x \) is equally distributed in both arms of the U-tube.
4. **Newton's Second Law:**
Applying Newton's second law:
\[ F = ma \]
\[ -2\rho g A x = \rho l a \]
Here, \( m = \rho l \) is the mass of the liquid, where \( l \) is the effective length of the liquid column (not the length of the tube).
5. **Simplifying the Equation:**
\[ a = -\frac{2g A}{l} x \]
Since \( a = \frac{d^2x}{dt^2} \):
\[ \frac{d^2x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c95aabf-9eb4-4d63-86ef-475a7c27077f%2Fb97fa7eb-ddb6-4059-80a8-71f11e362055%2Flezrnxc_processed.jpeg&w=3840&q=75)
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