Is Linear Momentum Conserved in a Collision Between a Disk and Particle?

In summary, the problem concerns the conservation of angular and linear momentum in a system consisting of a uniform disk and a particle colliding with the disk and getting lodged into it, causing the disk to spin. The conservation of angular momentum is used to find the angular velocity of the system after the collision, and the conservation of linear momentum is analyzed by considering the linear velocity of the center of mass of the system. The problem is solved by considering the force exerted by the fixed axis on the disk and how it dissipates the momentum and energy of the system.
  • #1
Dschumanji
153
1
I'm not exactly sure what is wrong with my analysis for this problem concerning the conservation of angular and linear momentum.

Problem Statement:
Suppose you have a uniform disk of mass M and radius R that can rotate about its central axis. A particle with mass M and velocity V strikes the rim of the disk (along a path tangent to the disk), gets lodged into it, and causes the disk to spin. Show that the linear momentum of the system is conserved.Attempted Solution:
I start by using the conservation of angular momentum:

MVR = (0.5MR^2 + MR^2)ω

The left side of the equation is the total angular momentum before the collision (it is just the angular momentum of the particle since the disk is stationary) and the right side is the angular momentum after the collision. The expression in parenthesis is the moment of inertia of the disk-particle system. With a bit of algebra you can conclude that:

ω = (2V)/(3R)

To analyze the linear momentum after the collision I look at the linear velocity of the center of mass of the disk-particle system. The center of mass is in between the center of the disk and the lodged particle, so the linear momentum is given by:

(0.5Rω)(2M) = (V/3)(2M) = (2/3)MV

However, the linear momentum before the collision is MV. Does anyone know what I am doing wrong?
 
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  • #2
The problem is if the disk rotates around a fixed axis or it is a free disk the particle is colliding with. If the disk rotates about a fix axis the linear momentum is not conserved as the axis exerts force.
Try to solve the problem with free disk. Then the whole system will rotate about the common centre of mass.

ehild
 
  • #3
The axis is fixed for this problem. Are you saying that the missing momentum is going into whatever is holding the axis fixed? If that is the case, problem solved! :p

Perhaps it is important to say that the disk is spinning with respect to the axis through the center of the disk (like a merry go round, not a coin spinning on its side) and not through the center of mass of the combined disk-particle system.
 
Last edited:
  • #4
You know that the change of linear momentum is equal to the impulse of the force exerted. FΔt=Δ(Ʃmv). The axis exerts force, but you can imagine that it "takes on" momentum when the particle collides with the disk (it bends a bit) and then starts to vibrate and dissipates the momentum and energy at the end.

ehild
 
  • #5
Ah, I see now! Thanks, Ehild! :biggrin:
 
  • #6
Welcome! (the green grin is very nice!):wink:

ehild
 

Related to Is Linear Momentum Conserved in a Collision Between a Disk and Particle?

1. What is missing linear momentum?

Missing linear momentum refers to the situation where the total linear momentum of a system is not conserved. This means that the sum of the momentums of all objects in the system is not equal to the initial total momentum. It can occur due to external forces acting on the system or due to miscalculations or errors in measurements.

2. How is linear momentum calculated?

Linear momentum is calculated by multiplying an object's mass by its velocity. In equations, it is represented as p = mv, where p is momentum, m is mass, and v is velocity. This calculation assumes that the object is moving in a straight line at a constant speed.

3. What factors can cause missing linear momentum?

There are several factors that can contribute to missing linear momentum. These include external forces acting on the system, such as friction or air resistance, miscalculations or errors in measurements, and the presence of hidden or unaccounted objects in the system.

4. How can missing linear momentum be corrected?

To correct for missing linear momentum, scientists must carefully analyze the forces acting on the system and ensure that all objects and their momentums are accounted for. Any errors in measurements or calculations should be identified and corrected. In some cases, external forces can be reduced or eliminated to conserve linear momentum.

5. Why is the conservation of linear momentum important?

The conservation of linear momentum is an important principle in physics because it helps us understand and predict the behavior of objects in motion. It also allows us to make accurate calculations and predictions in various fields, such as engineering, mechanics, and astrophysics. Additionally, the conservation of linear momentum is a fundamental law of nature that holds true in all physical systems and interactions.

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