Mechanical vibrations: colliding blocks

In summary, the conversation discusses how to calculate the period and amplitude of an oscillating system after a collision, where both masses are connected to a spring. The formula for the period is known, but the amplitude is still being determined. The conversation explores the damping coefficient and its effect on the oscillation, and also mentions that the masses should separate after collision, not stick together. The negative value of the root is explained as being an underdamped oscillation.
  • #1
vxr
25
2
Homework Statement
One block of mass ##m = 0.20 kg## traveling at velocity ##v = 5 m/s## collides elastically with a second block of mass ##M = 0.80 kg## resting on a frictionless surface and connected to a spring with elastic force constant ##k = 80 N/m##.
What is the angular velocity ##\omega##, period ##T##, and the amplitude ##A## of block’s oscillations? Determine the respective values for the case when damping effects will appear with overall damping coefficient ##\beta = 21/s##.
Relevant Equations
##\omega = \sqrt{\omega_{0}^2 - \beta^2}##
I saw this general formula:

##w_{0} = \sqrt{\frac{k}{m}}##

In my case both masses after collision create connected system, so ##w_{0} = \sqrt{\frac{k}{m+M}}##

Plugging it into ##\omega = \sqrt{\omega_{0}^2 - \beta^2}## gives :

##\omega = \sqrt{\frac{k}{m+M} - \beta^2} = \sqrt{80 - 21^2} < 0##

It's a root of negative value. Why is that happening? Am I doing something wrong, or perhaps the damping coefficient is relatively too large in this task?

One more question: I know how to calculate the period ##T##, but what about amplitude ##A##? Is it simply: ##A = A_{0}e^{-\beta t} \quad \land \quad A_{0} = 0 \Longrightarrow A = 0##? If this is the case, why is this ##A_{0} = 0## true?
 
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  • #2
The statement "collides elastically" usually means that the masses separate after the collision, not that they stick together. Treat the collision as occurring instantaneously with momentum and kinetic energy being conserved to find an expression for the velocity of the mass connected to the spring immediately after the collision.
 
  • #3
vxr said:
It's a root of negative value. Why is that happening?
@kuruman is right that you should only have used the mass attached to the spring, but that is not the explanation for the negative value. Clearly you could change the question to make it a coalescing collision.
The value is negative when it is an underdamped oscillation. See
http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html
 

FAQ: Mechanical vibrations: colliding blocks

What are mechanical vibrations?

Mechanical vibrations are a type of motion that occurs when an object oscillates back and forth around a fixed point. This motion is caused by external forces acting on the object, such as collisions or impacts.

How do colliding blocks produce mechanical vibrations?

Colliding blocks produce mechanical vibrations by transferring energy to each other during the collision. This energy causes the blocks to vibrate back and forth until the energy is dissipated.

What factors affect the intensity of mechanical vibrations in colliding blocks?

The intensity of mechanical vibrations in colliding blocks can be affected by various factors such as the mass, velocity, and elasticity of the blocks, as well as the angle and surface of impact.

How are mechanical vibrations measured?

Mechanical vibrations can be measured using a variety of instruments, such as accelerometers and seismographs, which detect and record the frequency, amplitude, and other characteristics of the vibrations.

What are some real-world applications of studying mechanical vibrations in colliding blocks?

Studying mechanical vibrations in colliding blocks has practical applications in fields such as engineering, seismology, and material science. It can help improve the design and performance of structures, predict and prevent earthquakes, and understand the properties of different materials.

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