Conservation of Angular Momentum: angular speed

In summary, the student is trying to find the new angular speed when the dumbbells are at 0.3m from the rotation axis. Using the homework equations, they find that the new angular speed is 0.750 rad/sec.
  • #1
mickellowery
69
0

Homework Statement


A student sits on a freely rotating stool holding 2 3.00kg dumbbells. When the students arms are extended horizontally the dumbbells are 1.00m from the axis of rotation and the student rotates with an angular speed of 0.750 rad/sec. The moment of inertia of the student and the stool is 3.00 kgm2 and it is assumed to be constant. The student pulls the dumbbells in to a position 0.300m from the rotation axis. What is the new angular speed?


Homework Equations


I was going to try Ii[tex]\omega[/tex]i=If[tex]\omega[/tex]f but the problem with this is that the problem says I is constant at 3.00 kgm2. I assume that I need to factor in the radius somehow but I'm not sure how to do it.


The Attempt at a Solution

 
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  • #2
When the dumbells are at 1m, what is the moment of inertia of them?

Add that to the inertia of the stool and you have the initial moment of inertia.

initial angular momentum = Iinitialωinitial.


When the dumbells are at 0.3m, what is the moment of inertia then? (Add this to get inertia of the stool to get the final moment of inertia)
 
  • #3
So can I model the dumbbells as a rod and use 1/12 ML2 as the moment of inertia or do I have to integrate [tex]\int[/tex][tex]\rho[/tex]dV? I am facing problems with both ways, I don't know what the volume would be if I integrate, and it seems like modeling as a rod would not be correct.
 
  • #4
mickellowery said:
So can I model the dumbbells as a rod and use 1/12 ML2 as the moment of inertia or do I have to integrate [tex]\int[/tex][tex]\rho[/tex]dV? I am facing problems with both ways, I don't know what the volume would be if I integrate, and it seems like modeling as a rod would not be correct.

No need to do all of that. Just treat them as point masses such that I=mr2
 
  • #5

To solve this problem, we can use the conservation of angular momentum equation, L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular speed. In this case, we will use the initial and final states of the student and stool system to solve for the final angular speed.

Initial state:
Ii = 3.00 kgm^2
ωi = 0.750 rad/sec

Final state:
Ii = 3.00 kgm^2
ωf = unknown
r = 0.300m

Using the conservation of angular momentum equation, we can set the initial and final angular momentums equal to each other:

Iiωi = Ifωf

Solving for ωf, we get:
ωf = (Iiωi)/If

Substituting in the values from the initial and final states, we get:
ωf = (3.00 kgm^2 * 0.750 rad/sec)/(3.00 kgm^2)

Simplifying, we get:
ωf = 0.750 rad/sec

Therefore, the new angular speed is the same as the initial angular speed, 0.750 rad/sec. This is because the moment of inertia and the angular momentum are constant, and the change in the radius does not affect the final angular speed. This is known as the conservation of angular momentum, where the total angular momentum of a system remains constant unless acted upon by an external torque.
 

1. What is conservation of angular momentum?

The conservation of angular momentum is a fundamental physical law that states the total angular momentum of a system remains constant unless acted upon by an external torque. This means that in a closed system, the total amount of angular momentum remains the same regardless of any internal changes or interactions.

2. How is angular momentum calculated?

Angular momentum is calculated by multiplying the moment of inertia of an object by its angular velocity. The moment of inertia is a measure of how difficult it is to change the rotational motion of an object, while angular velocity is a measure of how fast an object is rotating around a fixed axis.

3. What is meant by angular speed?

Angular speed is the rate at which an object rotates around a fixed axis. It is measured in radians per second and is related to linear speed by the radius of the object's circular path. The higher the angular speed, the faster the object is rotating.

4. How does conservation of angular momentum apply to real-world situations?

The conservation of angular momentum applies to a wide range of real-world situations, from the rotation of planets and galaxies to the motion of spinning toys. For example, a figure skater can increase their angular speed by pulling in their arms, thus decreasing their moment of inertia, while keeping their angular momentum constant.

5. What is the relationship between torque and conservation of angular momentum?

Torque is the force that causes an object to rotate around an axis. According to the conservation of angular momentum, the total amount of angular momentum in a system remains constant, so if an external torque is applied to an object, it will cause a change in the object's angular velocity to maintain the total angular momentum of the system.

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