Law of the Conservation of Angular Momentum > Principle of Conservation of Momentum

In summary, the conversation discusses applying a force to a lever to move an object, and how the force, distance, and time affect the momentum and kinetic energy of the system. It is found that the conservation of angular momentum and energy is at play, and that the direction of the force can affect the output momentum. The formulas for kinetic energy and momentum are also discussed, showing the consistency with physical knowledge. The conversation ends with a reminder that the conservation of momentum is actually the conservation of angular momentum, and that a lever with little to no mass would not have any momentum or kinetic energy.
  • #1
kmarinas86
979
1
We will apply a force of some distance for some period of time.

Let's say the goal is to apply a maximum force over a small distance.

We apply 1 Newton at right angles to a frictionless lever, 1 meter from the pivot point.
At 10 centimeters from that pivot point is an object that weighs 1 kg, and the force it receives is 10 Newtons. Because the force is 10 Newtons, the object gains 10 kg*m/s of momentum per second. Two special situations may be considered:
1) Had the force been higher, the object would gain greater than 10 kg*m/s due to greater acceleration.
2) For the same force, a heavier mass would accelerate more slowly but it would still absorb momentum at the same rate as determined by that same force.

The force on each lever arm is inversely proportional to the radius of each lever arm. Therefore, it can be assumed that this system as currently revealed involves the conservation of angular momentum. It also conserves energy. If one increases the length of the arm of input, the effort needed to be applied to do the same amount of work directly decreases.

The question then is, will the momentum going into the lever equal the momentum going out of the lever? If the force going into the lever is 1 Newton, it is absorbing momentum at a rate of 1 kg*m/s per second. If the force going out of the lever is 10 Newtons, it is applying momentum at a rate of 10 kg*m/s per second. This defies common sense. It implies that a rate of momentum utilization of 9 kg*m/s per second is unaccounted for. The question however is, what is the direction of that force? That force is anti-parallel to the force applied on that lever. If we ignore the lever itself and simply accounted for the input and output forces (which are in opposite directions), common sense tells us that a rate of momentum of 9 kg*m/s is being utilized from whatever medium to which the force of 1 Newton is applied and from which the force of 10 Newtons comes from. The momentum, it seems, would have to come from the lever itself. Incongruous!

Even more shocking is when we consider the Newtons applied over a period of time. A force times time equals a change in momentum. Apparently, the kinetic energy stored in some mass would increase with the square of the force and the square of the time the force is applied, assuming a constant force and mass. For a given untethered mass, we could apply twice the force for half the time or half the force for twice the time and it would result in the same kinetic energy gained. The question is however, how far does the lever arm move in that time period. In the case of a lever arm of twice the length, half the force is applied resulting in the same acceleration of the object with the same torque, and the force is made at twice the velocity, resulting in the same power input (since force*velocity=power; e.g. prony brake). The opposite case occurs for a lever arm half as long with an applied force twice as much. Again, moving the same mass at the same rate does not affect the input power or input torque.

However, starting with the same lever arm, the input force * the input arm radius / the output arm radius = the output force. We return to the conundrum we found where output momentum exceeded input momentum, suggesting that the momentum came from the lever itself. If we think the input lever arm radius * input force as the calculation for torque, and if we recognize that torque is quantified as angular momentum transferred per unit time, we can easily see that angular momentum is conserved.

KE = 1/2 * mv^2

where
KE = Kinetic Energy
m=mass
v=velocity

p = mv

where
p = momentum
m=mass
v=velocity

KE = 1/2 * p^2/m

where
p^2/m = mv^2

F = p/t

where
t=time
p=F*t

KE = 1/2 * (F*t)^2/m

where
KE = F*d
d=1/2 * (F/m)*t^2
(F/m)=acceleration
d=1/2 * a * t^2
v_initial=0
d=change in position

These formulas are consistent with physical knowledge and with the above paragraphs.

Lesson: Conservation of momentum is really the conservation of ANGULAR momentum.
 
Last edited:
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  • #2
kmarinas86 said:
The question then is, will the momentum going into the lever equal the momentum going out of the lever?
Not necessarily. The lever could be (nearly) massless. In that case it would have no momentum or kinetic energy at all.

After a quick scan, the things you said are correct. But it is so much harder to understand and error prone in verbal form rather than equation form.
 

1. What is the Law of Conservation of Angular Momentum?

The Law of Conservation of Angular Momentum states that the total angular momentum of a system remains constant as long as there is no external torque acting on the system. This means that the angular momentum of a system cannot be created or destroyed, it can only be transferred between objects.

2. How is the Law of Conservation of Angular Momentum related to the Principle of Conservation of Momentum?

The Law of Conservation of Angular Momentum is a specific case of the broader Principle of Conservation of Momentum. The Principle of Conservation of Momentum states that the total momentum of a system remains constant as long as there is no external force acting on the system. The Law of Conservation of Angular Momentum is a specialized version of this principle that applies specifically to rotational motion.

3. Why is the Law of Conservation of Angular Momentum important?

The Law of Conservation of Angular Momentum is important because it helps us understand and predict the behavior of rotating systems. It is a fundamental law of physics that applies to a wide range of phenomena, from the rotation of planets and galaxies to the spinning of a top or a figure skater on ice.

4. What are some real-life examples of the Law of Conservation of Angular Momentum?

Some examples of the Law of Conservation of Angular Momentum in action include the rotation of the Earth on its axis, the orbit of the Moon around the Earth, and the spinning of a bicycle wheel. In each of these cases, the angular momentum of the system remains constant and is not affected by external forces (ignoring the slight effects of friction).

5. Can the Law of Conservation of Angular Momentum be violated?

No, the Law of Conservation of Angular Momentum is a fundamental law of physics that has been observed and tested extensively through experiments. It has been found to hold true in all cases where there is no external torque acting on the system. However, it is important to note that it may appear to be violated in certain situations due to the presence of external forces or the transfer of angular momentum between objects.

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