I would really appreciate the help for this assignment. A small reasoning for the answer would be very helpful. Thank you.
A sphere of mass m and radius r is attached to the rim of a disk (mass M, radius R). The disk is rotating counter-clockwise with a constant period T around an axis through the center of the disk. Neglect friction. Consider the directions A through I at the indicated position of the sphere.
1. What is the direction of the external torque on the disk?
2. What is the direction of the net force on the sphere?
3. What is the direction of the acceleration of the sphere?
4. If the disk has a radius R=0.5 m and a mass M=0.9 kg, and the sphere has a radius r=0.02 m and a mass m=0.28 kg, and if the period of rotation is T=5 s, what is the magnitude of the angular velocity of the disk?
5. What is the magnitude of the velocity of the sphere?
6. What is the moment of inertia of the whole contraption around the axis through the center of the disk?
I (disk,cm) = (1/2)MR^2 I(sphere,cm) = (2/5)mr^2
7. Now, the disk is accelerated counter-clockwise to spin faster. What is the direction of the external torque?
8. While the disk is spun up, what is the direction of the acceleration of the sphere?
9. The sphere was not glued on very well and disconnects. In what direction is it going to fly off?
1. The direction of the external torque on the disk can be determined using the right-hand rule for rotational motion. Place your right hand with fingers curled in the direction of rotation of the disk. The thumb will then point in the direction of the external torque on the disk.
2. The net force on the sphere can be determined by analyzing the forces acting on it. Since there is no friction considered, the only force acting on the sphere is the tension in the string connecting it to the disk. The direction of the net force on the sphere is along the string, towards the center of the disk.
3. The direction of the acceleration of the sphere can be determined by considering the net force acting on it. Since the net force is towards the center of the disk, the acceleration of the sphere will also be towards the center of the disk.
4. To calculate the magnitude of the angular velocity of the disk, we can use the formula: angular velocity (ω) = 2π / period (T). Substituting the given values, ω = 2π / 5 s.
5. The magnitude of the velocity of the sphere can be found using the formula: velocity (v) = ω * r, where ω is the angular velocity of the disk and r is the radius of the sphere. Substituting the given values, v = (2π / 5 s) * 0.02 m.
6. The total moment of inertia of the system can be calculated by adding the moments of inertia of the disk and the sphere. Using the given formulas, the moment of inertia of the whole contraption is (1/2)MR^2 + (2/5)mr^2.
7. When the disk is accelerated counter-clockwise to spin faster, the direction of the external torque remains the same. The external torque will still be in the direction determined earlier (using the right-hand rule), which is opposite to the direction of the rotational motion.
8. While the disk is being spun up, the acceleration of the sphere will still be towards the center of the disk. The net force acting on the sphere (tension in the string) remains the same, resulting in the same direction of acceleration.
9. When the sphere disconnects from the contraption, it will fly off in a direction tangential to its previous circular path. This is because there is no longer any force acting on it to keep it moving in a circular path.