Determine angular velocity - Dynamics

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Homework Help Overview

The problem involves determining the angular velocity of a bar released from a horizontal position and passing through a vertical position, utilizing the principles of dynamics and conservation of energy. The context includes a torsional spring and gravitational potential energy considerations.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the application of conservation of energy, identifying types of energy involved, including elastic potential energy and gravitational potential energy. There are attempts to clarify the correct expressions for these energies and their roles in the problem.

Discussion Status

Participants are actively engaging with the problem, questioning assumptions about energy calculations and the behavior of the center of mass. Some guidance has been offered regarding the setup of energy equations and the importance of measuring height changes correctly.

Contextual Notes

There is ongoing confusion regarding the calculation of gravitational potential energy and the height of the center of mass as the bar transitions from horizontal to vertical. Participants are also discussing the moment of inertia relevant to the problem.

dvep
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Homework Statement



Torsional spring constant k=33 Nm/rad
It is undeflected when it is in the vertical position, it weighs 6.4 kg
When the bar is released from the horizontal position from rest, determine the angular velocity \omega as it passess the vertical position.
We can assume that it is frictionless.

The Attempt at a Solution



Am I right to say that I have to use the conservation of energy and find

T1+V1 = T2+V2 which would be my angualr velocity?

I am quite stuck on this question, I would be grateful for some guidance.
 

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Yes, you'll need to apply conservation of energy. Hint: There are three 'types' of energy involved here. What are they? What are expressions for each?
 
Doc Al said:
Yes, you'll need to apply conservation of energy. Hint: There are three 'types' of energy involved here. What are they? What are expressions for each?

Thank you for your reply

Elastic potential energy = 1/2kx^2
Rotational kinetic energy = 1/2Iw^2 = 1/2I(v/r)^2

Though I am unsure what the third energy is.

So,

EPE=1/2*33*(0.8/sin45)^2 = 20 J

Am I on the right track?

Sorry to be a pain, I haven't been taught this yet in class.
 
dvep said:
Thank you for your reply

Elastic potential energy = 1/2kx^2
OK, but for a torsion spring, the displacement from equilibrium will be in radians.
Rotational kinetic energy = 1/2Iw^2 = 1/2I(v/r)^2
OK.

Though I am unsure what the third energy is.
Hint: The bar has weight.

So,

EPE=1/2*33*(0.8/sin45)^2 = 20 J
Not sure what you did here. What angle (in radians) does the initial position make with the unstretched position?
 
doc al said:
ok, but for a torsion spring, the displacement from equilibrium will be in radians.

Oh yeah, so

EPE=1/2k\theta^2

doc al said:
hint: The bar has weight.

I wasn't sure but I was thinking gravitational potential energy?

doc al said:
not sure what you did here. What angle (in radians) does the initial position make with the unstretched position?

EPE=1/2*33*1.571^2 J
 
dvep said:
Oh yeah, so

EPE=1/2k\theta^2
Right.

I wasn't sure but I was thinking gravitational potential energy?
Exactly.

EPE=1/2*33*1.571^2 J
Good.
 
Doc Al said:
Right.


Exactly.


Good.

So,

EPE=40.723 J
GPE=50.176 J

Am I meant to add these together to get the total energy, then at the vertical position they both equal 0 and rotational kinetic energy equals the total energy, then use the rotational kinetic energy formula and rearrange to find the angualr velocity w?
 
dvep said:
EPE=40.723 J
OK.
GPE=50.176 J
How did you calculate this? For what position? Initial or final?

Am I meant to add these together to get the total energy, then at the vertical position they both equal 0 and rotational kinetic energy equals the total energy, then use the rotational kinetic energy formula and rearrange to find the angualr velocity w?
To avoid confusion, set up your conservation equation like so:

EPE1 + GPE1 + RotKE1 = EPE2 + GPE2 + RotKE2

(Some terms will end up being zero.)
 
Doc Al said:
How did you calculate this? For what position? Initial or final?

I thought that might be wrong

GPE1=mglcos(theta)= 6.4*9.8*0.8*cos(1.571) = 50.157 J
GPE2=mglcos(theta)= 6.4*9.8*0.8*cos(0) = 50.157 J

Is that correct, they are both the same.

Doc Al said:
To avoid confusion, set up your conservation equation like so:

EPE1 + GPE1 + RotKE1 = EPE2 + GPE2 + RotKE2

(Some terms will end up being zero.)

So what I get,

EPE1 + GPE1 + RotKE1 = EPE2 + GPE2 + RotKE2
40.723 + 50.157 + 0 = 0 + 50.157 + 40.723

RotKE2= 1/2Iw^2 = 1/2I(v/r)^2
Angular velocity = w = sqrt(2*RotKE2/mr^2) = sqrt(2*40.723/6.4*0.8^2) = 4.459 rads/s

Am I at least a little bit close?
 
  • #10
dvep said:
I thought that might be wrong

GPE1=mglcos(theta)= 6.4*9.8*0.8*cos(1.571) = 50.157 J
GPE2=mglcos(theta)= 6.4*9.8*0.8*cos(0) = 50.157 J

Is that correct, they are both the same.
No, that makes no sense. The bar rises from horizontal to vertical, so its GPE increases. Hint: Measure height from the surface. Hint 2: Consider what happens to the center of mass.
 
  • #11
Doc Al said:
No, that makes no sense. The bar rises from horizontal to vertical, so its GPE increases. Hint: Measure height from the surface. Hint 2: Consider what happens to the center of mass.

I'm still lost. What do you mean measure the height from the surface? Will the center of mass move towards the end of the bar?
 
  • #12
dvep said:
I'm still lost. What do you mean measure the height from the surface? Will the center of mass move towards the end of the bar?
When the bar goes from horizontal to vertical, what happens to its center of mass? That determines the change in gravitational PE.
 
  • #13
Doc Al said:
When the bar goes from horizontal to vertical, what happens to its center of mass? That determines the change in gravitational PE.

The GPE is 0 at the initial position and 50.157 J when vertical?
The center of mass gains translation kinetic energy?
 
  • #14
dvep said:
The GPE is 0 at the initial position
OK.
and 50.157 J when vertical?
No. How high does the center of mass rise?
The center of mass gains translation kinetic energy?
Sure, but you need not worry about that. Treat the bar as being in pure rotation about the axis, so all its kinetic energy is rotational KE about that axis.
 
  • #15
Doc Al said:
OK.

No. How high does the center of mass rise?

0.4 m?

so,

EPE1 + GPE1 + RotKE1 = EPE2 + GPE2 + RotKE2
40.723 + 0 + 0 = 0 + 25.088 + 15.635
 
  • #16
dvep said:
0.4 m?
Right.

so,

EPE1 + GPE1 + RotKE1 = EPE2 + GPE2 + RotKE2
40.723 + 0 + 0 = 0 + 25.088 + 15.635
OK, but since you are solving for RotKE2, I'd write it as:
EPE1 + GPE1 + RotKE1 = EPE2 + GPE2 + RotKE2
40.723 + 0 + 0 = 0 + 25.088 + RotKE2

Then you'd deduce that RotKE2 = 15.635

And then use RotKE = 1/2Iw^2 to find w.
 
  • #17
Doc Al said:
Right.


OK, but since you are solving for RotKE2, I'd write it as:
EPE1 + GPE1 + RotKE1 = EPE2 + GPE2 + RotKE2
40.723 + 0 + 0 = 0 + 25.088 + RotKE2

Then you'd deduce that RotKE2 = 15.635

And then use RotKE = 1/2Iw^2 to find w.

Thank you for your help.
 
  • #18
Doc Al said:
Right.

Then you'd deduce that RotKE2 = 15.635

And then use RotKE = 1/2Iw^2 to find w.

Just looked through this problem... to calculate the moment of inertia, I, do you use I = 1/3mL^2?
 
  • #19
Skittles999 said:
Just looked through this problem... to calculate the moment of inertia, I, do you use I = 1/3mL^2?
Yes.
 

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