A cylinder with moment of inertia 22.1 kg m2 rotates with angular velocity 5.71 rad/s on a frictionless vertical axle. A second cylinder, with moment of inertia 28.3 kgm2, initially not rotating, drops onto the first cylinder and remains in contact. Since the surfaces are rough, the two eventually reach the same an- gular velocity.
To determine the final angular velocity when two cylinders come into contact, we need to apply the principle of conservation of angular momentum. Angular momentum is conserved when no external torques act on a system.
The initial angular momentum of the first cylinder is given by:
L1 = I1 * ω1
Where:
L1 = initial angular momentum of the first cylinder
I1 = moment of inertia of the first cylinder
ω1 = initial angular velocity of the first cylinder
Similarly, the initial angular momentum of the second cylinder is zero as it is initially not rotating:
L2 = I2 * ω2
L2 = 0
Where:
L2 = initial angular momentum of the second cylinder
I2 = moment of inertia of the second cylinder
ω2 = initial angular velocity of the second cylinder
After the two cylinders come into contact and reach a common angular velocity ω, their combined angular momentum is:
L_combined = (I1 + I2) * ω
Since angular momentum is conserved, L_combined is equal to the sum of the initial angular momenta:
L_combined = L1 + L2
Therefore:
(I1 + I2) * ω = I1 * ω1
Rearrange the equation to solve for ω:
ω = (I1 * ω1) / (I1 + I2)
Now we can substitute the given values:
I1 = 22.1 kg m^2 (moment of inertia of the first cylinder)
ω1 = 5.71 rad/s (initial angular velocity of the first cylinder)
I2 = 28.3 kg m^2 (moment of inertia of the second cylinder)
Substituting these values into the equation, we have:
ω = (22.1 kg m^2 * 5.71 rad/s) / (22.1 kg m^2 + 28.3 kg m^2)
Now perform the calculations to find the final angular velocity.
To find the final angular velocity of the system, we can use the principle of conservation of angular momentum. According to this principle, the total angular momentum before the cylinders come into contact must be equal to the total angular momentum after they reach the same angular velocity.
The angular momentum of an object can be calculated using the formula:
L = I * ω
Where:
L = angular momentum
I = moment of inertia
ω = angular velocity
Let's calculate the initial and final angular momenta of the two cylinders separately and set them equal to each other.
For the first cylinder:
Initial angular momentum = I1 * ω1
For the second cylinder:
Initial angular momentum = I2 * 0 (as it is initially not rotating)
After the cylinders come into contact and reach the same angular velocity:
Final angular momentum = (I1 + I2) * ωf
Using the principle of conservation of angular momentum, we can set the initial angular momentum of the first cylinder equal to the final angular momentum of the combined system:
I1 * ω1 = (I1 + I2) * ωf
Now, let's substitute the given values:
I1 = 22.1 kg m^2 (moment of inertia of the first cylinder)
I2 = 28.3 kg m^2 (moment of inertia of the second cylinder)
ω1 = 5.71 rad/s (initial angular velocity of the first cylinder)
ωf = unknown (final angular velocity of the combined system)
22.1 kg m^2 * 5.71 rad/s = (22.1 kg m^2 + 28.3 kg m^2) * ωf
126.191 kg m^2/s = 50.4 kg m^2 * ωf
Dividing both sides by 50.4 kg m^2:
ωf = 126.191 kg m^2/s / 50.4 kg m^2
ωf = 2.501 rad/s (approximately)
Therefore, the final angular velocity of the combined system is approximately 2.501 rad/s.