Final answer:
No sand needs to fall on the disk to decrease the speed to 2.0 rev/s.
Step-by-step explanation:
To calculate how much sand must fall on the disk to decrease the speed to 2.0 rev/s, we can use the principle of conservation of angular momentum. The initial angular momentum of the disk is equal to the final angular momentum of the disk plus the angular momentum of the sand ring. Initially, the disk has an angular momentum of I1ω1, and finally, it will have an angular momentum of I2ω2, where I1 and ω1 are the initial moment of inertia and angular velocity of the disk, and I2 and ω2 are the final moment of inertia and angular velocity of the disk, respectively.
First, we need to find the initial moment of inertia (I1) and angular velocity (ω1) of the disk. The moment of inertia of the disk (I1) can be calculated using the formula I1 = (1/2)MR2, where M is the mass of the disk and R is the radius of the disk. Plugging in the values, we get I1 = (1/2)(0.3 kg)(0.3 m)2 = 0.027 kg·m2. The angular velocity (ω1) is given as 3.0 rev/s.
To calculate the final moment of inertia (I2) and angular velocity (ω2) of the disk, we need to take into account the added sand ring. Since the sand falls at a distance of 20 cm from the center, the radius of the sand ring is 20 cm. The moment of inertia of the sand ring (Ir) can be calculated using the formula Ir = mR, where m is the mass of the sand and R is the radius of the sand ring. The mass of the sand can be calculated by subtracting the mass of the disk from the total mass that falls on the disk. The total mass that falls on the disk can be calculated using the formula m_total = (I2 - I1)/R, where I2 is the final moment of inertia of the disk. Plugging in the values, we get m_total = (I2 - 0.027 kg·m2)/(0.2 m) = 0.3 kg. So, the mass of the sand is 0.3 kg - 0.3 kg = 0 kg, as all the sand falls on the disk. Therefore, the moment of inertia of the sand ring is Ir = 0 kg·m2.
Now, we can find the final moment of inertia (I2) by adding the moment of inertia of the disk (I1) and the moment of inertia of the sand ring (Ir). Therefore, I2 = I1 + Ir = 0.027 kg·m2 + 0 kg·m2 = 0.027 kg·m2. Finally, we can calculate the final angular velocity (ω2) using the equation I1ω1 = I2ω2. Plugging in the values, we get (0.027 kg·m2)(3.0 rev/s) = (0.027 kg·m2)(ω2), which gives us ω2 = 3.0 rev/s.
Now, we can calculate the amount of sand that needs to fall on the disk to decrease the speed to 2.0 rev/s. To find the mass of the sand needed, we can use the formula m_needed = (I2 - I1)/R, where I2 is the final moment of inertia of the disk, I1 is the initial moment of inertia of the disk, and R is the radius of the sand ring. Plugging in the values, we get m_needed = (0.027 kg·m2 - 0.027 kg·m2)/(0.2 m) = 0 kg. Therefore, no sand needs to fall on the disk to decrease the speed to 2.0 rev/s.