Question

A rigid disk, of homogeneous material, mass equal to 1 kg and radius 50 cm, is kept spinning on the floor without friction around an axis with constant angular velocity. A small ball, much smaller in size than the disk and with a mass equal to one-twentieth of the disk, is initially at a radial distance from the center of the disk equal to half the radius. After 10 s the ball arrives at the end of the disk and exits.

1. To what forces is the ball subjected in the reference system integral with the floor?
2. To what forces is the ball subjected in the reference system integral with the disk and having its
origin at the center of it?
3. What kind of motion does the ball have in both of the above reference systems?
4. Calculate the rotational velocity of the disc.
5. How, if at all, would the motion of the disk vary once the ball leaves it,

assuming it is no longer kept in constant rotation? (Type of motion and angular velocity
angular)?



Un disco rigido, di materiale omogeneo, massa pari a 1 Kg e raggio 50 cm, è mantenuto in rotazione sul pavimento senza attrito attorno ad un asse con velocità angolare costante. Una pallina, di dimensioni molto più piccole del disco e massa pari a un ventesimo del disco, si trova inizialmente ad una distanza radiale dal centro del disco pari alla metà del raggio. Dopo 10 s la pallina arriva all'estremità del disco e ne fuoriesce. 1. A quali forze è soggetta la pallina nel sistema di riferimento solidale con il pavimento?
2. A quali forze è soggetta la pallina nel sistema di riferimento solidale con il disco e avente origine al centro dello stesso?
3. Che tipo di moto ha la pallina in entrambi i sopracitati sistemi di riferimento?
4. Calcolare la velocità di rotazione del disco.
5. Come varierebbe, se varierebbe il moto del disco una volta che la pallina lo abbandona, nell'ipotesi che non sia più mantenuto in rotazione costante? (Tipo di moto e velocità angolare)?​

Answer 1

In the reference system integral with the floor, the ball is subjected to the force of gravity pulling it downwards towards the floor and a normal force pushing it upwards to balance the force of gravity. The ball is also subjected to the centrifugal force pushing it outwards away from the center of the disk.

In the reference system integral with the disk and having its origin at the center of it, the ball is not subjected to any net force. The centrifugal force is balanced by a centripetal force pulling it towards the center of the disk.

In both reference systems, the ball has a combination of circular motion and linear motion. In the reference system integral with the floor, the ball is moving in a circular path due to the centrifugal force and is also moving away from the center of the disk. In the reference system integral with the disk, the ball is moving in a circular path with a constant speed.

To calculate the rotational velocity of the disc, we can use the formula v = r * w, where v is the rotational velocity, r is the radius of the disk and w is the angular velocity. Therefore, the rotational velocity of the disk is v = 0.5 m * w = 25m/s

Once the ball leaves the disk, the disk will no longer be balanced and will lose angular momentum, so it will slow down and eventually stop spinning. The motion of the disk will change from rotational motion to translational motion. The angular velocity of the disk will decrease to zero.