Finding motion where the acceleration depends on position and time

Click For Summary

Discussion Overview

The discussion revolves around finding the motion of a system where the acceleration is dependent on both position and time. Participants are exploring how to formulate a differential equation using initial values and a specific acceleration expression, which is relevant to a pendulum modeled with a spring-like rope.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents the acceleration equation a(t) and seeks guidance on how to compute motion given initial conditions.
  • Another participant inquires about the familiarity with setting up and solving differential equations.
  • A participant suggests using Hamilton's principle as a potential approach to the problem.
  • Some participants discuss the possibility of resolving the problem component-wise, focusing on the x and y components of acceleration.
  • There is a suggestion to clarify the physical scenario, with participants agreeing that it represents a pendulum with a spring-like rope.
  • One participant questions the appropriateness of not using standard equations of motion, suggesting that the problem may be more advanced than initially indicated.
  • Another participant points out the need for clarity in defining vectors and suggests resolving forces along the spring and tangentially.
  • A comment is made regarding the definitions of acceleration and velocity, indicating a potential misunderstanding in the title of the thread.

Areas of Agreement / Disagreement

Participants express varying levels of agreement on the physical scenario and the approach to solving the problem. However, there is no consensus on the best method to compute the motion, and multiple viewpoints on the appropriateness of using certain equations remain unresolved.

Contextual Notes

There are limitations regarding the assumptions made about the system, the dependence on definitions of motion, and the complexity of integrating the acceleration function due to its dependence on the position vector.

simo22
Messages
4
Reaction score
0
I have computed that the acceleration in my problem is
a(t) = -gj - k/m(|r(t)| - L_0) * r(t)/|r(t)|

Where a(t) is the acceleration vector, g is the gravitational acceleration, j is the unit vector in y-direction, k is the spring constant, m is the mass, r(t) is the position vector, |r(t)| is the length of r(t) and L_0 is the equilibrium length of a rope. I am asked in my problem to make a differential equation by using some initial values and the acceleration to compute the motion.

I was told by the teacher not to use the equations of motion because those require a constant acceleration, while the one we have here is dependent on the position. Is there a way for me to compute the motion given an initial velocity and position?
 
Last edited by a moderator:
Physics news on Phys.org
Are you familiar with setting up and solving differential equations?
 
I'm somewhat familiar with it. I was thinking that since we know r(0), v(0) and a(t), I could simply compute
r(t) = r(0) + v(0)*t + ∫∫a(t)dtdt, but I do not know how to integrate a(t) when it depends on r(t)
 
simo22 said:
Summary:: Finding motion by making a differential equation using initial values and an acceleration that depends on position and time.

I have computed that the acceleration in my problem is
a(t) = -gj - k/m(|r(t)| - L_0) * r(t)/|r(t)|

Where a(t) is the acceleration vector, g is the gravitational acceleration, j is the unit vector in y-direction, k is the spring constant, m is the mass, r(t) is the position vector, |r(t)| is the length of r(t) and L_0 is the equilibrium length of a rope. I am asked in my problem to make a differential equation by using some initial values and the acceleration to compute the motion.

I was told by the teacher not to use the equations of motion because those require a constant acceleration, while the one we have here is dependent on the position. Is there a way for me to compute the motion given an initial velocity and position?

What physical scenario does this represent?
 
It is a pendulum where we think of the rope as a spring
 
simo22 said:
It is a pendulum where we think of the rope as a spring

Ah, ##r(t)## is a vector! You must write ##\vec r##. And the thing on the end is ##\hat r##. I guess you mean the bob is attached by a spring. It can't be a rope.

So, how are you going to resolve your vectors? ##\hat x, \hat y## or along the line of the spring and tangential to it?
 
Once more I'd rather suggest to use Hamilton's principle (which is adequate for an I-level thread, I hope).
 
Well, I have also computed the components of the acceleration, a_x and a_y, so I guess I could do it component-wise
 
simo22 said:
Well, I have also computed the components of the acceleration, a_x and a_y, so I guess I could do it component-wise

At this level, you may struggle without using Latex to render your equations:

https://www.physicsforums.com/help/latexhelp/

Another observation: this seems quite an advanced problem, so the comment about "not using equations of motion", by which you mean "not using SUVAT" looks strange. If you are looking at this problem, then the simplicity of SUVAT should be long ago! Have you not studied simple harmonic motion or other variable force problems before this?

In this case, perhaps resolving along the spring and tangential to it seems more natural. I haven't got time to analyse this myself, but my guess is that's a more promising approach.
 
  • #10
2nd derivative of distance with respect to time= acceleration. your title seems to have pointed to the definition of velocity, which is the 1st derivative of distance with respect to time. acceleration is the change of velocity with respect to time
 

Similar threads

High School Constant jerk motion: acceleration vs. position
  • · Replies 2 ·
Replies
2
Views
2K
High School Sign conventions in torque and non-uniform circular motion
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Finding the potential energy if force depends on both position and time
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad How can I calculate the time for non-uniform acceleration in a circular path?
  • · Replies 13 ·
Replies
13
Views
3K
Graduate Motion of a spring that has mass
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Non-Newtonian description of an accelerated object
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Non-Constant Angular Acceleration
  • · Replies 6 ·
Replies
6
Views
3K
High School Maximum velocity for projectile motion?
  • · Replies 25 ·
Replies
25
Views
4K
Undergrad Normal modes of vibration from the total energy
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Projectile Motion: Path's length?
  • · Replies 18 ·
Replies
18
Views
3K