A turntable with a moment of inertia of 0.017 kg*m^2 rotates feely at 3.3 rad/s. A circular disk of mass 240 g and diameter 26 cm, and initially not rotating, slips down a spindle and lands on the turntable.
Angular momentum is conserved.
I1*w1 = (I1 + I2)w2
I1 = 0.017 kgm^2
I2 = (1/2)M2*R^2
w1 = 3.3 rad/s
R = 0.13 m
Solve for w2.
By the way, what was your question?
To determine the final angular velocity of the system after the disk lands on the turntable, we need to apply the principle of conservation of angular momentum.
Angular momentum (L) is defined as the product of moment of inertia (I) and angular velocity (ω):
L = I * ω
In this case, the initial angular momentum L_initial of the turntable is given by:
L_initial = I_turntable * ω_turntable
The final angular momentum L_final of the system (turntable + disk) can be calculated by adding the angular momenta of the turntable and the disk:
L_final = I_turntable * ω_final + I_disk * ω_disk
where ω_final is the final angular velocity of the system and I_disk is the moment of inertia of the disk.
Since the disk initially is not rotating, its initial angular velocity (ω_disk) is zero.
Now, let's calculate the moment of inertia for the disk. The moment of inertia of a solid disk can be calculated using the formula:
I_disk = (1/2) * m_disk * r^2
where m_disk is the mass of the disk and r is the radius of the disk.
Given that the mass of the disk (m_disk) is 240 g (0.240 kg) and the diameter of the disk is 26 cm (0.26 m), the radius (r) of the disk is half the diameter:
r = 0.26 m / 2 = 0.13 m
Plugging in the values, we can calculate the moment of inertia of the disk:
I_disk = (1/2) * 0.240 kg * (0.13 m)^2 = 0.002988 kg*m^2
Now, let's substitute the values into the conservation of angular momentum equation:
L_initial = I_turntable * ω_turntable
L_final = I_turntable * ω_final + I_disk * ω_disk
Since angular momentum is conserved, L_initial = L_final:
I_turntable * ω_turntable = I_turntable * ω_final + I_disk * ω_disk
Rearranging the equation to solve for ω_final:
ω_final = (I_turntable * ω_turntable - I_disk * ω_disk) / I_turntable
Plugging in the known values:
ω_final = (0.017 kg*m^2 * 3.3 rad/s - 0.002988 kg*m^2 * 0 rad/s) / 0.017 kg*m^2
Calculating:
ω_final = 3.3 rad/s
Therefore, the final angular velocity of the system after the disk lands on the turntable is 3.3 rad/s.