A uniform disk with a mass of 800 g and radius 17.0 cm is rotating on frictionless bearings with an rotational speed of 18.0 Hz when Jill drops a 130 g clod of clay on a point 7.10 cm from the center of the disk, where it sticks. What is the new rotational speed of the disk?
To find the new rotational speed of the disk after the clay is dropped and sticks to the disk, we can apply the principle of conservation of angular momentum.
The angular momentum of an object is the product of its moment of inertia (I) and its angular velocity (ω). The moment of inertia of a disk can be calculated using the formula: I = (1/2) * m * r^2, where m is the mass and r is the radius.
Initially, the disk is rotating with a certain angular velocity, given by ω1 = 18.0 Hz. The clay is dropped and sticks to the disk at a distance of 7.10 cm from the center, which can be considered as an increase in the moment of inertia. After the clay sticks, the combined system (disk + clay) will have a new moment of inertia, denoted as I'.
According to the conservation of angular momentum, the initial angular momentum (L1) should equal the final angular momentum (L2).
L1 = L2
The initial angular momentum of the disk is given by:
L1 = I * ω1
The final angular momentum of the combined system is given by:
L2 = I' * ω2, where ω2 is the new rotational speed we need to find.
Since the clay sticks to the disk and their angular velocities become the same, we can consider the moment of inertia of the combined system as the sum of the moment of inertia of the disk and the moment of inertia of the clay.
I' = I + m_clay * r_clay^2
Substituting the values into the equation:
I' = (1/2) * m_disk * r_disk^2 + m_clay * r_clay^2
= (1/2) * 800 g * (17.0 cm)^2 + 130 g * (7.10 cm)^2
Note: We first need to convert the mass and dimensions to SI units before calculating.
After finding the value of I', we can rearrange the conservation of angular momentum equation to solve for ω2:
ω2 = L1 / I'
Substituting the values of L1 and I' into the equation, we can calculate the new rotational speed of the disk.