Conservation of linear momentum and rotational motion

In summary, linear momentum is conserved in a closed system as long as there are no external forces present. In the case of a hinged door or rod, there is an external force from the hinge which contributes to the linear momentum and angular momentum of the system. When a ball collides with the door, its linear momentum is transferred to the door, causing it to rotate. This rotation conserves the door's linear momentum within the door-hinge system, while the angular momentum is conserved within the rotation itself. The hinge acts as a constraint for the door, and the linear momentum of the rotating door is best described using constraints rather than Newtonian mechanics.
  • #1
kavan
5
0
When you open a door you apply force in any particular direction and as a result you get rotational motion of the door. My question is how linear momentum is conserved in this case as linear momentum seems to have generated rotational motion? To clarify my question further, if we fix a rod from one end such that it can freely rotate about that end and then hit another end of the rod with a speeding ball with some linear momentum along any direction(say x).. the momentum will be transferred to the rod which will start to rotate...now the rotating rod will have linear momentum with components in both directions(say x and y). How did the momentum along the y direction come into picture when originaly there was none?8
 
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  • #2
kavan said:
When you open a door you apply force in any particular direction and as a result you get rotational motion of the door. My question is how linear momentum is conserved in this case

Linear momentum is conserved in a closed system as long as there are no external forces on the system. In the case of a hinged door or rod there is an external force from the hinge.

How did the momentum along the y direction come into picture when originaly there was none?

The force from the hinge has a component in the y direction.
 
  • #3
jbriggs444 said:
Linear momentum is conserved in a closed system as long as there are no external forces on the system. In the case of a hinged door or rod there is an external force from the hinge.
We can include the external force within the system by suitably expanding it. Imagine collision case, a ball hitting the door makes one system on which there are no external force.
jbriggs444 said:
The force from the hinge has a component in the y direction.
Didnt get that. If originally there were no momentum in y direction from where the momentum in y direction comes once the door starts rotating.
 
  • #4
The hinge connects the door to... what exactly? If you are expand the system to include the wall the hinge is screwed into then you have to include the motion of the wall in your calculations.
 
  • #5
Plz see the screenshot attached. How did linear momentum of the ball converted into angular momentum of the rotating road. I hope I've clarified my question.
 

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  • #6
kavan said:
How did linear momentum of the ball converted into angular momentum of the rotating road.
Even before the collision, the ball has angular momentum about the hinge. Angular momentum is conserved during the collision. (Linear momentum is not. As already pointed out, you cannot ignore the force of the hinge on the door.)
 
  • #7
jbriggs444 said:
The hinge connects the door to... what exactly? If you are expand the system to include the wall the hinge is screwed into then you have to include the motion of the wall in your calculations.
And the wall is attached to the building. And the building is attached to the earth.

Linear momentum is conserved in the door, wall, building, Earth system.

Now excuse me. I hear my Noether calling.
 
  • #8
As for converting linear to angular momentum you have to know that everything that has linear momentum has angular momentum too and the other way round.

When it comes to the collision the energy of the ball is given to the door as momentum. The only motion that the door can perform is a rotation and therefore all of the energy goes in it. During the rotation the door's linear momentum is conserved within the system door-hinge. The angular momentum is conserved for itself in the rotation.

When you calculate it you will see that the angular momentum of the ball in relation to a random point is the same as the angular momentum of the door related to the same point.

The linear momentum of the rotating door cannot be described so well using Newtonian mechanics. Instead you need the term of constraints since the hinge is a constraint for the door.
 

What is the conservation of linear momentum?

The conservation of linear momentum is a fundamental principle in physics that states that the total linear momentum of a closed system remains constant over time, unless an external force is applied to the system. This means that in a collision or other interaction between objects, the total momentum of the objects before and after the interaction will be the same.

How does the conservation of linear momentum apply to rotational motion?

In rotational motion, the conservation of linear momentum applies in a similar way, but with angular momentum instead. Angular momentum is the product of an object's moment of inertia and its angular velocity. Just as linear momentum is conserved in a closed system, so is angular momentum. This means that in a rotating system, the total angular momentum before and after an interaction will be the same, unless an external torque is applied.

What is the difference between linear and angular momentum?

Linear momentum refers to the motion of an object along a straight line, while angular momentum refers to the motion of an object around a fixed point. Linear momentum is a vector quantity, meaning it has both magnitude and direction, while angular momentum is a pseudovector, meaning it has magnitude but not direction. Additionally, linear momentum is conserved in all inertial reference frames, while angular momentum is only conserved in a fixed reference frame.

How is the conservation of linear momentum and rotational motion related to Newton's laws of motion?

The conservation of linear momentum and rotational motion are related to Newton's laws of motion through the principle of inertia. Newton's first law states that an object at rest will remain at rest, and an object in motion will continue in motion in a straight line, unless acted upon by an external force. This principle also applies to rotational motion, where an object will continue to rotate at a constant speed unless acted upon by an external torque.

What are some real-world applications of the conservation of linear momentum and rotational motion?

The conservation of linear momentum and rotational motion have many practical applications in our daily lives. Some examples include car accidents, where the principle of conservation of momentum can be used to determine the forces involved in a collision, and the motion of satellites and planets, where the conservation of angular momentum helps explain their orbits around larger objects. In sports such as gymnastics and figure skating, the conservation of angular momentum is also important for understanding the movements and rotations of athletes.

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