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.
a) Find the new angular speed.
b) what is the change in kinetic energy?
To find the new angular speed and the change in kinetic energy, we can apply the principles of conservation of angular momentum and conservation of kinetic energy.
a) Finding the new angular speed:
Since the initial angular momentum of the system is conserved, we can use the equation:
\(I_1 \cdot \omega_1 = I_2 \cdot \omega_2\)
where \(I_1\) and \(\omega_1\) represent the initial moment of inertia and angular speed respectively, and \(I_2\) and \(\omega_2\) represent the final moment of inertia and angular speed respectively.
Given:
\(I_1 = 0.017 \, \text{kg} \cdot \text{m}^2\) (initial moment of inertia)
\(I_2 = I_1 + \text{moment of inertia of the disk}\)
\(= I_1 + \frac{1}{2} \cdot M \cdot R^2\)
where \(M\) represents the mass of the disk and \(R\) represents the radius (half of the diameter of the disk).
\(M = 240 \, \text{g} = 0.24 \, \text{kg}\) (mass of the disk)
\(R = \frac{26 \, \text{cm}}{2} = 0.13 \, \text{m}\) (radius of the disk)
Substituting the given values into the equation, we can solve for \(\omega_2\):
\(0.017 \cdot \omega_1 = (0.017 + \frac{1}{2} \cdot 0.24 \cdot 0.13^2) \cdot \omega_2\)
\(3.3 = (0.017 + 0.0249) \cdot \omega_2\) [Substituting \(3.3\) rad/s for \(\omega_1\)]
\(3.3 = 0.0419 \cdot \omega_2\)
\(\omega_2 = \frac{3.3}{0.0419}\)
\(\omega_2 \approx 78.761 \, \text{rad/s}\)
Therefore, the new angular speed of the turntable is approximately \(78.761 \, \text{rad/s}\).
b) Finding the change in kinetic energy:
The change in kinetic energy can be calculated by subtracting the initial kinetic energy from the final kinetic energy.
The initial kinetic energy of the system is zero since the disk is initially not rotating.
The final kinetic energy can be calculated using the equation:
\(KE_2 = \frac{1}{2} \cdot I_2 \cdot \omega_2^2\)
where \(KE_2\) represents the final kinetic energy.
Substituting the given values into the equation, we can solve for \(KE_2\):
\(KE_2 = \frac{1}{2} \cdot (0.017 + 0.0249) \cdot (78.761)^2\)
\(KE_2 \approx 209.294 \, \text{J}\)
Therefore, the change in kinetic energy is approximately \(209.294 \, \text{J}\).