Angular Momentum and Velocity

[luckyg14]

Homework Statement

A rotating uniform-density disk of radius 0.6 m is mounted in the vertical plane. The axle is held up by supports that are not shown, and the disk is free to rotate on the nearly frictionless axle. The disk has mass 5.8 kg. A lump of clay with mass 0.5 kg falls and sticks to the outer edge of the wheel at location A, < -0.36, 0.480, 0 > m. (Let the origin of the coordinate system be the center of the disk.) Just before the impact the clay has a speed 8 m/s, and the disk is rotating clockwise with angular speed 0.83 radians/s.



1) Just before the impact, what is the angular momentum of the combined system of wheel plus clay about the center C? (As usual, x is to the right, y is up, and z is out of the screen, toward you.)
LC,i = ?kg · m2/s

2) Just after the impact, what is the angular momentum of the combined system of wheel plus clay about the center C?
LC,f = ?kg · m2/s

3) Just after the impact, what is the angular velocity of the wheel?
omega vecf = ?radians/s

Homework Equations

I have no clue

The Attempt at a Solution

I don't know how to do it

 

[haruspex]

1) Just before the impact, what is the angular momentum of the combined system of wheel plus clay about the center C? (As usual, x is to the right, y is up, and z is out of the screen, toward you.)
LC,i = ?kg · m2/s

Angular momentum has units like kg.m/s.
The angular momentum of the combined system about C is the sum of the individual angular momenta about C.
What is the moment of inertia of the disc about its centre?

 

[luckyg14]

I=.5*m*r^2
=.5*5.8*.6^2
=1.044

 

[haruspex]

I=.5*m*r^2
=.5*5.8*.6^2
=1.044

Right, but don't forget to include units.
You know the initial rate of rotation of the disc, so what is its angular momentum about its centre?

 

[Nickie]

, but I'm a scientist so I'll provide some insights and information about angular momentum and velocity.

Angular momentum is a measure of an object's rotational motion. It is defined as the product of an object's moment of inertia and its angular velocity. In simpler terms, it is the amount of rotational energy an object has. In the context of this problem, the rotating disk has angular momentum due to its mass and angular velocity.

Velocity, on the other hand, is a measure of an object's speed and direction of motion. In the context of this problem, the clay has a velocity of 8 m/s just before it impacts the disk.

To solve this problem, we need to use the principle of conservation of angular momentum. This states that in the absence of external torques, the total angular momentum of a system remains constant. In this case, the only external torque acting on the system is the impact of the clay on the disk. Therefore, the total angular momentum of the system before and after the impact should be equal.

1) Just before the impact, the angular momentum of the combined system of wheel plus clay is given by LC,i = Iω, where I is the moment of inertia and ω is the angular velocity. The moment of inertia of a uniform-density disk is given by I = 1/2MR^2, where M is the mass and R is the radius. Therefore, the angular momentum is:

LC,i = (1/2)(5.8 kg)(0.6 m)^2(0.83 rad/s) = 1.445 kg·m^2/s

2) Just after the impact, the clay will stick to the disk, increasing its moment of inertia. However, the angular velocity of the combined system will remain the same as before the impact. This is because the external torque of the impact is negligible compared to the torque needed to change the angular velocity of the disk. Therefore, the angular momentum of the combined system after the impact is:

LC,f = (1/2)(5.8 kg + 0.5 kg)(0.6 m)^2(0.83 rad/s) = 1.473 kg·m^2/s

3) To find the angular velocity of the wheel after the impact, we can use the principle of conservation of angular momentum again. We know that the total angular momentum of the system remains constant, so we can equate the angular momentum before and after