Solving for Angular Velocity and Rotation Angle in a Collision System

In summary, the problem is that there is a collision between two masses and the angular velocity of the system after the collision is 6M2*v/ (6M1L + 3M2L + ML).
  • #1
vu10758
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The problem is problem 6 at

http://viewmorepics.myspace.com/index.cfm?fuseaction=viewImage&friendID=128765607&imageID=1460003525

A long thin rod of mass M and length L with two balls of mass M1 (same mass for both) attached is allowed to rotate about the horizontal axis shown. The bar is initially stationary. It is then hig with a piece of putty of mass M2 and speed v which sticks to one of the M1's.

a) Find the angular velocity of the system after the collision. The correct answer should be 6M2*v/ (6M1L + 3M2L + ML)

b) What angle will the system rotate through before coming to a stop? Assume that it must be between 180 and 270 degrees.



For part a,

Am I suppose to use the conservation of angular momentum.

IW = I_f*W_f
(1/12)ML^2 *w= (1/12)(M1+M2)*L^2*W_f

I am stuck though since I don't know how to account for v.

For part b,

The answer is 180 + arcsin(V^2/gL)

I don't know how to get to v^2/gL.
 
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  • #2
The rod with attached masses has a moment of inertia you can calculate and no initial angular momentum. You account for v by looking up the fundamental definition of angular momentum (for a moving particle; not the derived expression involving moments of inertia for a rigid assumbly of particles). It involves the mass and velocity and what else?
 
  • #3
Angular momentum = m*r*v

I know that I for a rod is (1/12)ML^2

so L = Iw

then

L = (1/12)ML^2*v^2/L^2
L = (1/12)MV^2*L^2

r = 2L



If I have

IW = I_f*W_f

W_f = I/I_f * W

W_f = (1/12)ML^2/(I_f) * W

I am stuck. I don't know what to do from here.
 
  • #4
vu10758 said:
Angular momentum = m*r*v

I know that I for a rod is (1/12)ML^2

so L = Iw

then

L = (1/12)ML^2*v^2/L^2
L = (1/12)MV^2*L^2

r = 2L



If I have

IW = I_f*W_f

W_f = I/I_f * W

W_f = (1/12)ML^2/(I_f) * W

I am stuck. I don't know what to do from here.
You need to fix the relationships between r and L and between v and ω. Be careful to distinguish the initial particle velociy from the velocity after the collision, and be careful about the lengths involved in the problem.
 
  • #5
I made some mistakes. L = 2r, r = (1/2)L

w=v/r
w^2=v^2/r^2 or v^2/(1/2*L)^2 = 4v^2/L

Is M in this case 2M1 since we have two masses in the system before collision. Should the mass be 2M1 + M2 after collision?
 
  • #6
For conservation of the angular momentum of the interacting system:

[tex]L_{before}=L_{after}[/tex]

therefore

[tex]L_{putty}=L_{system}[/tex]

[tex]\frac{l}{2}p_{putty} = \left(I_{rod} + I_{m_1m_2} + I_{m_1}\right) \omega[/tex]
 
  • #7
Thanks. For part b, I still don't know where v^2/gl come from. G is an acceleration due to gravity, l is a length, and v^2 is m^2/s^2. After division, v^2/gl is just a number with no unit. However, I don't know where the term come from.
 
  • #8
Try and approach it along these lines

[tex]W_{torques} = \Delta K[/tex]

The system experiences two torques, [tex]\Gamma _1,\ \Gamma_{12}[/tex] which is from the weights of m1 and (m1 + m2). The torques will change as the system rotates so you need to integrate to find the work done by these.

The final kinetic energies are zero. So we are left with only the initial rotational kinetic energies of the three components of the system.
 
Last edited:
  • #9
Note that from part a that

[tex]\omega _i = \frac{pl}{2 I_s}[/tex]

where [tex]I_s[/tex] is the moment of inertia of the system

also note that the change in kinetic energy of the system will be

[tex]\Delta K = -\frac{1}{2} I_s {\omega _i}^2[/tex]
 
Last edited:
  • #10
Changed previous post.
 

What is angular velocity?

Angular velocity is a measure of how fast an object is rotating about a fixed axis. It is usually represented by the symbol ω (omega) and is measured in radians per second.

How do you calculate angular velocity?

Angular velocity can be calculated by dividing the change in angle (θ) by the change in time (t). It can be expressed as ω = θ/t.

What is the difference between angular velocity and linear velocity?

Angular velocity is a measure of how fast an object is rotating, while linear velocity is a measure of how fast an object is moving in a straight line. Angular velocity is measured in radians per second, while linear velocity is measured in meters per second.

What factors affect angular velocity?

The main factor that affects angular velocity is the radius of rotation. Objects with a larger radius will have a slower angular velocity compared to objects with a smaller radius. Other factors that may affect angular velocity include friction, mass, and applied torque.

How is angular velocity used in real-life applications?

Angular velocity is used in many real-life applications, such as in the design of engines, turbines, and other rotating machinery. It is also used in sports, such as in calculating the rotation of a gymnast in a routine. In astronomy, angular velocity is used to measure the rotation of planets and stars.

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