A device consists of eight balls each of mass 0.4 kg attached to the ends of low-mass spokes of length 1.8 m, so the radius of rotation of the balls is 0.9 m. The device is mounted in the vertical plane. The axle is held up by supports that are not shown, and the wheel is free to rotate on the nearly frictionless axle. A lump of clay with mass 0.13 kg falls and sticks to one of the balls at the location shown, when the spoke attached to that ball is at 45 degrees to the horizontal. Just before the impact the clay has a speed 8 m/s, and the wheel is rotating counterclockwise with angular speed 0.20 radians/s.
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(a) Which of the following statements are true about the device and the clay, for angular momentum relative to the axle of the device?
The angular momentum of the device is the same before and after the collision.Just before the collision the angular momentum of the wheel is 0.The angular momentum of the device is the sum of the angular momenta of all eight balls.The angular momentum of the device + clay just after the collision is equal to the angular momentum of the device + clay just before the collision.The angular momentum of the falling clay is zero because the clay is moving in a straight line.
(b) Just before the impact, what is the angular momentum of the combined system of device 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
(c) Just after the impact, what is the angular momentum of the combined system of device plus clay about the center C?
LC,f = ???kg · m2/s
(d) Just after the impact, what is the angular velocity of the device?
omega vecf = ???radians/s
To answer these questions, we need to apply the principle of conservation of angular momentum. Angular momentum is a vector quantity defined as the product of rotational inertia and angular velocity.
(a) Let's evaluate each statement one by one:
1. The angular momentum of the device is the same before and after the collision.
This statement is true because angular momentum is conserved if there are no external torques acting on the system. In this case, since there are no external torques acting on the device, its angular momentum remains constant.
2. Just before the collision, the angular momentum of the wheel is 0.
This statement is false. The wheel has angular momentum due to its rotational motion before the collision. However, the direction of this angular momentum will change after the collision due to the addition of angular momentum from the falling clay.
3. The angular momentum of the device is the sum of the angular momenta of all eight balls.
This statement is false. The device consists of multiple balls, but angular momentum is not simply the sum of individual angular momenta because they might have different directions. The angular momentum of the device is determined by the overall rotational motion of the system.
4. The angular momentum of the device + clay just after the collision is equal to the angular momentum of the device + clay just before the collision.
This statement is true. Conservation of angular momentum states that the total angular momentum of an isolated system remains constant, so the total angular momentum before the collision is equal to the total angular momentum after the collision.
5. The angular momentum of the falling clay is zero because the clay is moving in a straight line.
This statement is false. Angular momentum is not limited to rotational motion. Even though the clay is moving in a straight line, it still has angular momentum due to its rotational motion around the axle of the device before the collision.
(b) Just before the impact, the angular momentum of the combined system of the device plus clay about the center C can be calculated as the sum of the individual angular momenta of the device and the clay:
LC,i = L_device,i + L_clay,i
(c) Just after the impact, the angular momentum of the combined system of the device plus clay about the center C can be again calculated as the sum of the individual angular momenta of the device and the clay:
LC,f = L_device,f + L_clay,f
(d) Just after the impact, the angular velocity of the device can be determined using the equation:
LC,f = (I_device + I_clay) * omega_vecf
where I_device is the rotational inertia of the device and I_clay is the rotational inertia of the clay.
To obtain the specific numerical values for LC,i, LC,f, and omega_vecf, more information about the properties of the device, such as the distribution of mass and the masses of the individual balls, is required.