Angular Momentum of an object with respect to a moving point

In summary, the conversation discusses two different methods for calculating the angular momentum of particle B with respect to A about a fixed point O. The first method involves subtracting the angular momenta of A and B about O, while the second method involves considering the relative velocity of B with respect to A and using the position vectors to calculate the angular momentum. It is determined that the first method is incorrect and the second method is the correct approach.
  • #1
Tanya Sharma
1,540
135

Homework Statement



Two particles A and B having equal masses m are rotating around a fixed point O with constant angular speed ω .A is connected to point O with a string of length L/2 whereas B is connected to point A with string of length L/2 .Find the angular momentum of B with respect to A about O.

O-------L/2-------A-------L/2--------B

Homework Equations





The Attempt at a Solution



There can be two approaches

1.We find angular momentum of B w.r.t O ,say L1 = mω2L.Then we find angular momentum of A w.r.t O ,say L2 = (mω2L)/4.Since angular momentum is a vector , angular momentum of B w.r.t A should be vector difference of L1 and L2 =(3/4)(mω2L)

2.We find relative velocity of B w.r.t A =ωL/2 ,i.e we have considered particle A to be at rest .Then we find shortest distance between point A and line of motion of B which is L/2 .
Now,angular momentum of A w.r.t B =(mω2L)/4

I feel approach 1 is the correct way of calculating angular momentum of a point with respect to a moving point ,but the correct answer is the one given by approach 2.

Which is the right way ?
 
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  • #2
Hello.

To see that method 1 doesn’t work, consider the case where the equal masses A and B are at the ends and the axis O is in the middle: A----------O----------B. The system revolves about O. Then A and B would have the same angular momentum vector about O. So, if you tried to get the angular momentum of B relative to A by subtracting the angular momenta of A and B about O, you would get zero. But that can’t be correct as B is moving relative to A.

For a general situation, you can see more formally why it doesn’t work by noting that ##\vec{L}_{A/O} =m_A \vec{r}_{A/O} \times\vec{v}_{A/O}##. Similarly for particle B. If you subtract them, you can see that it's not possible in general to reduce it to ##\vec{L}_{B/A} =m_B \vec{r}_{B/A} \times\vec{v}_{B/A}##

The reason you can subtract two velocity vectors to get a relative velocity is because of the relation between the position vectors: ##\vec{r}_{B/A} = \vec{r}_{B/O} - \vec{r}_{A/O}## which holds at each instant of time. Taking the time derivative gives the relative velocity formula.
 
  • #3
Hello TSny

Thank you very much for the explanation :smile:

So , I guess my misconception stemmed from the fact that I was treating Angular Momentum which is a vector in the same manner as we treat position vectors .ie [itex] \vec{L}_{A/B} =\vec{L}_{A/O}+\vec{L}_{B/O} [/itex] ,which is not the correct way. Instead I should have dealt Angular Momentum of A w.r.t as [itex]\vec{L}_{B/A} =m_B \vec{r}_{B/A} \times\vec{v}_{B/A}[/itex] .

Am I correct in assessing my mistake ?
 
  • #4
Yes, I think that's right. There's no reason why angular momentum vectors would be related the same as position vectors.
 
  • #5
TSny...thanks once again
 

1. What is Angular Momentum?

Angular momentum is a measure of the rotational motion of an object around a fixed point or axis. It is calculated by multiplying the moment of inertia of the object by its angular velocity.

2. What is the difference between Angular Momentum and Linear Momentum?

Linear momentum is a measure of the straight line motion of an object, while angular momentum is a measure of the rotational motion of an object. Linear momentum is calculated by multiplying an object's mass by its velocity, while angular momentum is calculated by multiplying an object's moment of inertia by its angular velocity.

3. How is Angular Momentum affected by the position of the moving point?

The position of the moving point has a significant effect on the angular momentum of an object. If the object is closer to the moving point, the angular momentum will be greater. Conversely, if the object is farther away from the moving point, the angular momentum will be smaller.

4. Can Angular Momentum be conserved?

Yes, angular momentum can be conserved in a closed system where no external forces act on the system. This is known as the law of conservation of angular momentum.

5. How is Angular Momentum related to Torque?

Angular momentum and torque are closely related as they are both measures of rotational motion. Torque is the force that causes rotation, while angular momentum is the measure of an object's rotational motion. In other words, torque is the cause of angular momentum.

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