# Visualizing the SHM of 2 blocks attached by a spring

• yucheng
In summary, The two blocks, a and b, slide on a straight track without friction. Block a is hit suddenly and has an instantaneous velocity to the right. The velocity of block b at later times is calculated using the equations. The graphs show simple harmonic motion and displacement towards the right.
yucheng
From Kleppner's Intro to Mechanics (Example 4.7, wording not exact): Two identical blocks a and b each of mass m slide without friction on a straight track. They are attached by a spring with unstretched length l and spring constant k; the mass of the spring is negligible compared to the mass of the blocks. Initially the system is at rest. At t = 0, block a is hit sharply, giving it an instantaneous velocity v 0 to the right. Find the velocity of each block at later times. (Try this yourself if there is a linear air track available—the motion is unexpected.)

After doing some Mathematics, we arrive at the formulas:

Let ##\omega=\sqrt{\frac{k}{m}}##
$$v_{a}=\frac{v_{0}}{2}(1+\cos{\omega t})$$
$$v_{b}=\frac{v_{0}}{2}(1-\cos{\omega t})$$

Question 1: Does anyone have a video/animation that shows such an oscillation? I have tried searching the web but to no avail.

Question 2: Is there a general strategy to visualize such motions based on the equations? Graphs are obviously criptic: just a bunch of lines. Is there a systematic way to understand them?

I have tried using Mathematica to plot the graphs, by fixing arbitrary but still reasonable values for ##v_0=6## and ##\omega=2##:

Graphs of the velocity:

Which shows simple harmonic motion as expected.

$$x_a = \frac{6}{2} (t + 2 \sin{2 t})$$
$$x_b = \frac{6}{2} (t - 2 \sin{2 t})$$

The displacement graph? (I got it by integrating the above equations of the velocity)

At leats it shows displacement towards the right, presumably more positive y.

Is this method correct? Obviously I am missing out on quite a few details: I am struggling to imagine the actual motion!

Last edited:
Delta2
Life becomes a bit more transparent in this case if you look at ##v_a+v_b## and at ##v_a-v_b## :
You 'see' motion of the center of mass and the spring oscillation

yucheng

Last edited:

## 1. What is SHM and how does it relate to the motion of 2 blocks attached by a spring?

SHM stands for Simple Harmonic Motion, which is a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position. In the case of 2 blocks attached by a spring, the blocks will oscillate back and forth due to the force of the spring pulling them towards the equilibrium position.

## 2. How can the SHM of 2 blocks attached by a spring be visualized?

The SHM of 2 blocks attached by a spring can be visualized by plotting the displacement of the blocks over time. This will result in a sinusoidal wave, with the amplitude representing the maximum displacement of the blocks and the period representing the time it takes for one complete oscillation.

## 3. What factors affect the SHM of 2 blocks attached by a spring?

The SHM of 2 blocks attached by a spring is affected by several factors, including the mass of the blocks, the stiffness of the spring, and the initial conditions (e.g. initial displacement and velocity). These factors can change the amplitude, period, and frequency of the oscillations.

## 4. How does the energy of the system change during SHM of 2 blocks attached by a spring?

The total energy of the system (kinetic + potential) remains constant during SHM of 2 blocks attached by a spring. As the blocks oscillate, their kinetic energy increases while their potential energy decreases, and vice versa. However, the total energy remains the same as long as there is no external force acting on the system.

## 5. Can the SHM of 2 blocks attached by a spring be used to model real-world systems?

Yes, the SHM of 2 blocks attached by a spring can be used to model many real-world systems, such as pendulums, springs in car suspensions, and even the motion of atoms in a crystal lattice. By understanding the principles of SHM, scientists can better understand and predict the behavior of these systems.

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