Two disks are originally separated. The mass of Disk #1 is 1.5 kg, and the mass of disk #2
is 2.6 kg. Disk #1 has a radius of 10 cm, and it is rotating at an angular velocity of 15.0
rad/sec. Disk #2 has a radius of 15 cm, and it rotates in the opposite direction at an
angular velocity of 22.0 rad/sec. These two disks are then joined together such that there
is no slippage between them. That is their mutual rotational velocity in rad/sec?
To find the mutual rotational velocity of the two disks after being joined together, we can use the principle of conservation of angular momentum.
The formula for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
To find the total angular momentum of the system before joining, we can calculate the individual angular momenta of Disk #1 and Disk #2.
The moment of inertia I of a disk is given by the formula I = (1/2)mr^2, where m is the mass and r is the radius.
For Disk #1:
Mass (m1) = 1.5 kg
Radius (r1) = 0.10 m
Angular velocity (ω1) = 15.0 rad/sec
I1 = (1/2)(1.5 kg)(0.10 m)^2 = 0.0075 kg·m^2
Angular momentum of Disk #1 (L1) = I1ω1 = (0.0075 kg·m^2)(15.0 rad/sec) = 0.1125 kg·m^2·rad/sec
For Disk #2:
Mass (m2) = 2.6 kg
Radius (r2) = 0.15 m
Angular velocity (ω2) = -22.0 rad/sec (negative because it rotates in the opposite direction)
I2 = (1/2)(2.6 kg)(0.15 m)^2 = 0.0585 kg·m^2
Angular momentum of Disk #2 (L2) = I2ω2 = (0.0585 kg·m^2)(-22.0 rad/sec) = -1.287 kg·m^2·rad/sec
Now, since the disks are joined together without slippage, the total angular momentum before joining is equal to the total angular momentum after joining.
Total angular momentum before joining = Total angular momentum after joining
L1 + L2 = L_total
0.1125 kg·m^2·rad/sec + (-1.287 kg·m^2·rad/sec) = L_total
Simplifying:
-1.1745 kg·m^2·rad/sec = L_total
Therefore, the mutual rotational velocity of the two disks after being joined together is -1.1745 rad/sec.
To find the mutual rotational velocity of the two disks when they are joined together, you can use the principle of conservation of angular momentum.
The angular momentum of an object is defined as the product of its moment of inertia and its angular velocity. The moment of inertia is a measure of an object's resistance to changes in its rotational motion and depends on both the mass and the distribution of mass around the axis of rotation.
In this case, we have two disks with different masses and radii. The moment of inertia of a disk is given by the formula:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass of the disk, and r is the radius of the disk.
For Disk #1:
m1 = 1.5 kg
r1 = 0.1 m
w1 = 15.0 rad/sec
Using the formula for moment of inertia, we can calculate the moment of inertia of Disk #1:
I1 = (1/2) * m1 * r1^2 = (1/2) * 1.5 kg * (0.1 m)^2 = 0.0075 kg*m^2
For Disk #2:
m2 = 2.6 kg
r2 = 0.15 m
w2 = -22.0 rad/sec (negative because it is rotating in the opposite direction)
Using the same formula, we can calculate the moment of inertia of Disk #2:
I2 = (1/2) * m2 * r2^2 = (1/2) * 2.6 kg * (0.15 m)^2 = 0.0585 kg*m^2
Since angular momentum is conserved, the sum of the angular momenta of the two disks before they are joined is equal to the angular momentum of the combined system after they are joined.
L1 + L2 = (I1 * w1) + (I2 * w2)
where L1 and L2 are the angular momenta of Disk #1 and Disk #2, and w1 and w2 are their respective angular velocities.
L1 + L2 = (0.0075 kg*m^2 * 15.0 rad/sec) + (0.0585 kg*m^2 * -22.0 rad/sec)
L1 + L2 = 0.1125 kg*m^2 * rad/sec - 1.287 kg*m^2 * rad/sec
L1 + L2 = -1.1745 kg*m^2 * rad/sec
After the two disks are joined, the total moment of inertia can be found by adding the individual moments of inertia.
I = I1 + I2 = 0.0075 kg*m^2 + 0.0585 kg*m^2 = 0.066 kg*m^2
The final angular velocity is then given by:
w = (L1 + L2) / I = (-1.1745 kg*m^2 * rad/sec) / 0.066 kg*m^2
w = -17.7818 rad/sec (negative because the direction of rotation is opposite)
Therefore, the mutual rotational velocity of the two disks when they are joined together is approximately -17.78 rad/sec.