# Question about two falling masses connected with a spring

In summary, the differential equation to solve is: m\frac{d^2z}{dt} + 2kz = 0 Where z is the distance of the bottom mass from the center of mass of the system. The solution the author posted in the above-mentioned topic doesn't seem coherent to me from a mathematical point of view. Any guidance would be appretiated.
Hi, this is my first post :)

I have a question in reference to the same problem in: https://www.physicsforums.com/showthread.php?t=67697&highlight=Falling+Mass

I am having trouble comprehending the 3rd part.

I understand that the differential equation to solve is:
$$m\frac{d^2y}{dt} + 2ky = mg$$

Solving this equation yields the following complete solution:
$$z(t) = \frac{mg}{2k} + c_1cos(\sqrt{2k/m}t) + c_2sin(\sqrt{2k/m}t)$$

I am stuck at this point. The solution the author posted in the above-mentioned topic doesn't seem coherent to me from a mathematical point of view. Any guidance would be appretiated.

Hi, this is my first post :)

I have a question in reference to the same problem in: https://www.physicsforums.com/showthread.php?t=67697&highlight=Falling+Mass

I am having trouble comprehending the 3rd part.

I understand that the differential equation to solve is:
$$m\frac{d^2y}{dt} + 2ky = mg$$
It appears to me that the differential equation should be:
$$m\frac{d^2z}{dt} + 2kz = 0$$

Where z is the distance of the bottom mass from the center of mass of the system.

The acceleration of the bottom mass is (mg-kx)/m and the acceleration of the entire system is g. So the relative acceleration is (mg-kx)/m - g
=-kx/m = -2kz/m (since z=1/2 the total extension x). a=-2kz/m leads to the diff. eq above.

Complete solution is (as you posted but without one term):
$$z(t)=c_1cos(\sqrt{2k/m}t) + c_2sin(\sqrt{2k/m}t)$$

Solving this with z(0)=mg/2k and z'(0)=0 gives the particular solution:

$$z(t)=(mg/2k) cos(\sqrt{2k/m}t)$$

Thank you for your reply. I can understand what is going on now. I was basically mixing up the acceleration relative to the ground with the acceleration relative to the center of mass.

## 1. What is the concept of "two falling masses connected with a spring"?

The concept of "two falling masses connected with a spring" is an example of a simple harmonic motion system. It involves two masses connected by a spring, where one mass is held stationary and the other is released from a certain height. The spring acts as a restoring force, causing the masses to oscillate back and forth until they eventually come to rest at their equilibrium position.

## 2. How do the masses affect the motion of the system?

In this system, the masses play a crucial role in determining the period and frequency of the oscillation. The larger the masses, the slower the oscillation, while smaller masses will result in a faster oscillation. Additionally, the masses also affect the amplitude of the oscillation, with larger masses resulting in a larger amplitude.

## 3. What is the role of the spring in this system?

The spring in this system acts as a restoring force, meaning it will always try to return to its original length when stretched or compressed. In the case of the two falling masses, the spring provides a force that brings the masses back to their equilibrium position, causing them to oscillate back and forth.

## 4. How does the height from which the masses are released affect the motion?

The height from which the masses are released will determine the initial potential energy of the system. The higher the masses are released, the more potential energy they will have, resulting in a larger amplitude of oscillation. However, this will not affect the period or frequency of the oscillation.

## 5. Can this system be used to model real-life situations?

Yes, the concept of "two falling masses connected with a spring" can be applied to real-life situations. For example, it can be used to model the motion of a mass on a spring, such as a pendulum or a car's suspension system. It can also be used to study the behavior of atoms and molecules in a solid or gas state.

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