Conservation of Energy and Momentum in Swinging Pendulum

[UMICHPremed]

Are energy and momentum conserved in a swinging pendulum (neglect air resistance)?

thanks!

According to the BR answer key they are not; however, I think ultimately while there is energy conversion from potential to kinetic in a cyclic manner, the total energy remains constant and so does momentum right?

Someone please explain!

---

Members don't see this ad.

 

[plsfoldthx]

At height of pendulum U = mgh momentum = mv = 0 (because no velocity)

A bottom of pendulum KE = .5mv^2 momentum = mv

hence energy is conserved, momentum is not.

 

[lorenzomicron]

Are energy and momentum conserved in a swinging pendulum (neglect air resistance)?

thanks!

According to the BR answer key they are not; however, I think ultimately while there is energy conversion from potential to kinetic in a cyclic manner, the total energy remains constant and so does momentum right?

Someone please explain!

Hey OP,

for a pendulum, the energy is conserved for sure. it just cycles between potential and kinetic.

as the poster above said, all the energy is kinetic at the bottom of the trajectory and all the energy is potential at the highest point of the trajectory.
this is because the gravitational force is a conservative field force.

however, the momentum of the pendulum is not conserved, because a condition for momentum conservation is that no external force should act of the system.

thus, momentum is not conserved because of gravity's action.

hope that makes it clear.

 

[UMICHPremed]

Thanks for the responses.

I understand that the basis for conservation of momentum is no external forces, and you're right, gravity is acting on the pendulum blob.

I guess ultimately what I am asking is the definition of conservation. Momentum is changing but when the bob from time(1) to time(2) is at the some position in its oscillation it will have the same momentum and ratio of kinetic energy to potential energy; thus, they are all "conserved" right.

But if conservation is defined as not changing and constant through out its motion, then momentum is not conserved, kinetic energy is not conserved, potential energy is not conserved, but total energy is conserved. What is the definition of conservation that should be used on the MCAT?

 

[lorenzomicron]

Thanks for the responses.

I understand that the basis for conservation of momentum is no external forces, and you're right, gravity is acting on the pendulum blob.

I guess ultimately what I am asking is the definition of conservation. Momentum is changing but when the bob from time(1) to time(2) is at the some position in its oscillation it will have the same momentum and ratio of kinetic energy to potential energy; thus, they are all "conserved" right.

But if conservation is defined as not changing and constant through out its motion, then momentum is not conserved, kinetic energy is not conserved, potential energy is not conserved, but total energy is conserved. What is the definition of conservation that should be used on the MCAT?

Conservation means the magnitude (i.e. if its a vector quantity like momentum then the root of the sum of the square of the separate components) must be constant and un-changing at all points in motion.

Momentum in this case is not conserved because it is changing with respect to t, x, and y.

Energy, however, is conserved. The key thing is that total energy E = K+U must be conserved, which it is as long as the force field is conservative, etc.

Potential energy conservation and kinetic energy conservation would be very specific limiting cases. For instance, in the case of linear momentum conservation from something like impact, kinetic energy is conserved (when there is no lost energy due to impact). I'm not sure but perhaps kinetic energy is conserved when momentum is conserved. Potential energy conservation just means there's no change in motion or inertia (think Newton's first law sorta).

But, what matters, is that total energy E is conserved. For a rigorous mathematical background of this, just see Noether's Theorem. hope this helps.

 

[dingyibvs]

Thanks for the responses.

I understand that the basis for conservation of momentum is no external forces, and you're right, gravity is acting on the pendulum blob.

I guess ultimately what I am asking is the definition of conservation. Momentum is changing but when the bob from time(1) to time(2) is at the some position in its oscillation it will have the same momentum and ratio of kinetic energy to potential energy; thus, they are all "conserved" right.

But if conservation is defined as not changing and constant through out its motion, then momentum is not conserved, kinetic energy is not conserved, potential energy is not conserved, but total energy is conserved. What is the definition of conservation that should be used on the MCAT?

Technically, the momentum is conserved, but then you'd have to include the Earth into the system. The pendulum itself is only one part of the system, and the momentum of individual parts of a system does not necessarily have to be conserved. Thus, the fact that the momentum within a pendulum is not conserved does not violate the conservation of momentum.

Momentum within a closed system is always, ALWAYS conserved, no matter what. The problem here is that the system includes both the Earth and the pendulum.

Also, to answer your question, yes, conservation = something being constant.

As for total energy, however, the system can be contained within the pendulum itself, so it is a constant even when you only look at the pendulum. The kinetic energy and the potential energy will trade back and forth so neither of those quantities is conserved.

 

[BerkReviewTeach]

Not to pile on another thought here, but whether total energy is conserved or not depends on whether the system is dampened or not. They state not to worry about air resistance, but that's doesn't mean that there is not friction at the hinge point.

For pendulum systems, you need to first consider whether energy is being lost (work is being done by friction). If energy is being lost then total energy is not conserved (and the bob doesn't reach as high of a maximum height on successive swings and it doesn't reach as great a maximum speed on successive swings). If there is no friction and the max height stays constant, then total energy is conserved.

Momentum of the pendulum is not conserved, because it is constantly changing during the flight of the pendulum. As you noted, gravity is acting on it. Dingyibvs brings up the fact that you need to pay attention to what exactly is included in the system you are considering. If it's just the pendulum, as your qurestion implies, then momentum is not conserved. But if it includes the earth, then as one gains momentum, the other counters it (recoils if you will), and momentum is conserved.

As far as the MCAT is concerned, it will likely be straight forward as described above, but do your due diligence and make sure it's just the pendulum bob they are asking about and whether friction is affecting its swing.

 

[lorenzomicron]

Technically, the momentum is conserved, but then you'd have to include the Earth into the system.

Momentum within a closed system is always, ALWAYS conserved, no matter what. The problem here is that the system includes both the Earth and the pendulum.

Are you sure about momentum of the earth being conserved? Technically, the earth is also subject to the larger gravitational field of the sun, thus as a system subject to an external force, its momentum is also not conserved. Thoughts?

 

[dingyibvs]

Are you sure about momentum of the earth being conserved? Technically, the earth is also subject to the larger gravitational field of the sun, thus as a system subject to an external force, its momentum is also not conserved. Thoughts?

Here's how you resolve that issue. Momentum is mass times velocity, but velocity, as we all know, is relative. When you consider the Earth-pendulum system, you're considering only their relative velocities(i.e. the pendulum swing), and since the sun pulls on the two systems pretty much equally, it does not really affect their velocities relative to each other and therefore does not change the conservation of momentum within the system.

 

[lorenzomicron]

Here's how you resolve that issue. Momentum is mass times velocity, but velocity, as we all know, is relative. When you consider the Earth-pendulum system, you're considering only their relative velocities(i.e. the pendulum swing), and since the sun pulls on the two systems pretty much equally, it does not really affect their velocities relative to each other and therefore does not change the conservation of momentum within the system.

So,then, in what manner could we calculate/say momentum is conserved in such a system? (Just trying to find out the math behind it.)

 

[dingyibvs]

So,then, in what manner could we calculate/say momentum is conserved in such a system? (Just trying to find out the math behind it.)

Which system? The earth-pendulum system? A geostationary satellite would probably be a good place to observe it, and you should, theoretically, see the pendulum move toward the earth as the earth moves toward the pendulum a tiny, tiny, little bit, and the exact opposite when the pendulum swings upward.