# Equation of motion: spring mass system - free undamped vibration

• jason.bourne
In summary, the conversation was about understanding the direction of inertia force in relation to the direction of motion and acceleration in a mechanical system. The participants discussed the use of D'Alembert's Principle and how to properly represent the inertia force in equations of motion. They also clarified the concept of defining the positive direction of acceleration and the role it plays in determining the direction of inertia force.
jason.bourne

## Homework Statement

i have uploaded my question. please check out the attached .pdf file.

## The Attempt at a Solution

#### Attachments

• doubt.pdf
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Shouldn't the 'inertial force' oppose the acceleration, not the motion?

i was following this website. here's the link

http://lpsa.swarthmore.edu/Systems/MechTranslating/TransMechSysModel.html

it has been mentioned in this website that 'inertia force acts opposite to the direction of motion'.

did i understand it right?
mass will decelerate in the positive x direction, when it moves away from equilibrium towards +A.
it will accelerate in the negative x direction, when it is moving towards the equilibrium position.
it will start decelerating in the negative x direction, when it moves away from equilibrium position towards -A.
it will start accelerating in the positive x direction, when it moves towards equilibrium position.

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jason.bourne said:
it has been mentioned in this website that 'inertia force acts opposite to the direction of motion'.

did i understand it right?
I think it's just a sloppy and misleading way to describe the direction. Note how they 'derive' the inertia force: It's just the negative of 'ma'.

Since your variable x already defines the positive direction of acceleration, $\ddot{x}$, the inertia force is simply $-m\ddot{x}$. The direction is already included: When the sign of $\ddot{x}$ changes, so will the sign of the inertia force.

But it's the direction of the acceleration that determines the direction of the inertia force, not the direction of motion.

Since your variable x already defines the positive direction of acceleration, x¨, the inertia force is simply −mx¨

as i defined downward x as positive, so does that mean x¨ is positive as i move downwards away from equilibrium?

i might be asking really very silly doubt but this thing is really confusing me. i want it to be sorted.

as the mass moves away from equilibrium towards +A in the downward positive x direction, if we draw a free body diagram, how do we represent acceleration?

if we take x¨, acceleration positive in the downward direction, the inertia force acts in the opposite direction i.e, upwards.

so if i write equation of motion using D'Alembert's Principle, i get:

-mx¨ - kx = 0.

but if i consider the situation where the mass is at a position, away from -A, towards equilibrium, then what is the direction of acceleration and how do i write the equation of motion?

Last edited by a moderator:
jason.bourne said:
as i defined downward x as positive, so does that mean x¨ is positive as i move downwards away from equilibrium?
$\ddot{x}$ is positive as long as the acceleration is in the positive direction, which you have defined as downward. Note that the restoring force of the spring always acts to accelerate the mass back towards the equilibrium point. So at any point below the equilibrium point, the acceleration would be upward.

i might be asking really very silly doubt but this thing is really confusing me. i want it to be sorted.

as the mass moves away from equilibrium towards +A in the downward positive x direction, if we draw a free body diagram, how do we represent acceleration?
You wouldn't put the acceleration on a free body diagram, but it would act upward.

if we take x¨, acceleration positive in the downward direction, the inertia force acts in the opposite direction i.e, upwards.
$\ddot{x}$ would be negative and the inertia force would be positive. But you don't have to worry about the inertia force direction--you'd still represent that inertia force as $-m\ddot{x}$.

so if i write equation of motion using D'Alembert's Principle, i get:

-mx¨ - kx = 0.
Right.

but if i consider the situation where the mass is at a position, away from -A, towards equilibrium, then what is the direction of acceleration and how do i write the equation of motion?
So now you are looking at the situation when the mass is above the equilibrium point. So the spring force acts downward and the acceleration is downward.

Nonetheless, you'd still write the restoring force as -kx (since x is now negative, the force comes out positive). And you'd still write the inertial force as opposite to the acceleration: $-m\ddot{x}$. (Since the acceleration will be positive, that will come out to be negative.)

So the equation of motion will be exactly the same. Realize that you don't have to know the direction of the acceleration ahead of time. Just represent the inertia force properly--with respect to your chosen coordinates. Then your equation of motion will tell you the direction of the acceleration.

thanks Doc Al. i think i got it.

if i have any doubts further, i'll get back to you.
thank you very much!

Excellent. Glad that it helped. And you are most welcome.

## What is the equation of motion for a spring mass system?

The equation of motion for a spring mass system is given by F = ma, where F is the force applied to the mass, m is the mass of the object, and a is the acceleration of the object.

## What is a free undamped vibration?

A free undamped vibration is the natural oscillation of a spring mass system without any external forces or damping present. This means that the system will continue to vibrate at its natural frequency indefinitely.

## How do you calculate the natural frequency of a spring mass system?

The natural frequency of a spring mass system can be calculated using the equation fn = √(k/m), where k is the spring constant and m is the mass of the object. This represents the frequency at which the system will naturally vibrate without any external forces.

## What is the amplitude of a spring mass system's vibration?

The amplitude of a spring mass system's vibration refers to the maximum displacement of the mass from its equilibrium position during one full cycle of oscillation. It is affected by factors such as the initial conditions and the energy applied to the system.

## What is the relationship between the spring constant and the natural frequency of a spring mass system?

The spring constant and the natural frequency of a spring mass system are directly proportional. This means that as the spring constant increases, the natural frequency of the system also increases. This can be seen in the equation fn = √(k/m), where a higher spring constant results in a higher natural frequency.

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