A turntable with a moment of inertia of 0.014 kg*m2 rotates feely at 2.7 rad/s. A circular disk of mass 500 g and diameter 22 cm, and initially not rotating, slips down a spindle and lands on the turntable.
a) Find the new angular speed.
rad/s
b) what is the change in kinetic energy?
J
To find the new angular speed and change in kinetic energy, we need to apply the principle of conservation of angular momentum and conservation of mechanical energy.
a) The initial angular momentum of the turntable (L1) is given by:
L1 = I1 * ω1
where I1 is the initial moment of inertia of the turntable and ω1 is the initial angular speed.
Substituting the given values:
L1 = 0.014 kg*m^2 * 2.7 rad/s
When the disk lands on the turntable, the moment of inertia changes, and the angular momentum is conserved. So we have:
L1 = L2
I1 * ω1 = I2 * ω2
where I2 is the moment of inertia of the turntable and the disk together, and ω2 is the new angular speed.
The moment of inertia of the disk (I_disk) can be calculated using the formula:
I_disk = 0.5 * m * r^2
where m is the mass of the disk and r is its radius (diameter / 2).
Substituting the given values for the disk:
I_disk = 0.5 * 0.5 kg * (0.22 m / 2)^2
Next, for the combined moment of inertia (I2), we have:
I2 = I1 + I_disk
Substituting the given values and the calculated value for I_disk into the equation, we can solve for ω2:
I1 * ω1 = (I1 + I_disk) * ω2
0.014 kg*m^2 * 2.7 rad/s = (0.014 kg*m^2 + 0.00242 kg*m^2) * ω2
Simplifying, we find:
ω2 = (0.014 kg*m^2 * 2.7 rad/s) / (0.014 kg*m^2 + 0.00242 kg*m^2)
Calculating ω2 will give us the new angular speed.
b) The change in kinetic energy can be calculated using the formula:
ΔKE = KE2 - KE1
where KE1 is the initial kinetic energy and KE2 is the final kinetic energy.
The initial kinetic energy (KE1) is given by:
KE1 = 0.5 * I1 * ω1^2
Substituting the given values for I1 and ω1, we can calculate KE1.
The final kinetic energy (KE2) is given by:
KE2 = 0.5 * I2 * ω2^2
Substituting the calculated values for I2 and ω2 into the equation will give us KE2.
Finally, subtracting KE1 from KE2 will give us the change in kinetic energy (ΔKE).
To summarize:
a) Calculate ω2 using the conservation of angular momentum principle.
b) Calculate KE1 and KE2, then find ΔKE by subtracting KE1 from KE2.