A bug that has a mass mb = 1 g walks from the center to the edge of a disk that is freely turning at 38 rpm. The disk has a mass of md = 11 g. If the radius of the disk is R = 30 cm, what is the new rate of spinning in rpm?
Angular momentum (I*w) is conserved. However the moment of inertia (I) increases and the angular velocity (w) decreases.
The initial moment of inertia is
I1 = (1/2)md*R^2 = 4950 g*cm^2
The final moment of inertia is
I2 = (1/2)md*R^2 + mb*R^2
= 4950 + 900 = 5850 g*cm^2
new rpm/old rpm = 4950/5850
To find the new rate of spinning in rpm, we need to apply the law of conservation of angular momentum.
The angular momentum of an object can be defined as the product of its moment of inertia and its angular velocity.
The moment of inertia of a rotating object depends on its mass distribution and the axis of rotation. For a disk rotating around an axis perpendicular to its surface (like in this case), the moment of inertia can be calculated by the formula:
I = (1/2) * m * R^2
where I is the moment of inertia, m is the mass of the object, and R is its radius.
For the bug:
mbug = 1 g = 0.001 kg
R = 30 cm = 0.3 m
Using the formula above, we can calculate the moment of inertia of the bug:
Ibug = (1/2) * mbug * R^2
Next, we need to calculate the initial angular momentum (Linitial) of the system before the bug moves to the edge of the disk. It can be defined as:
Linitial = Iinitial * winitial
where Iinitial is the initial moment of inertia of the system and winitial is the initial angular velocity of the disk.
For the disk:
mdisk = 11 g = 0.011 kg
R = 30 cm = 0.3 m
Using the formula for the moment of inertia of a disk, we can calculate the initial moment of inertia of the system:
Idisk = (1/2) * mdisk * R^2
Since the disk is freely turning, its initial angular velocity (winitial) is given as 38 rpm. To convert this to rad/s, we need to multiply it by 2π/60:
winitial = (38 rpm) * (2π/60)
Now we can calculate the initial angular momentum of the system:
Linitial = Idisk * winitial
After the bug moves to the edge of the disk, it will carry some angular momentum with it. Let's call this additional angular momentum carried by the bug Lbug.
Using the law of conservation of angular momentum, we can write:
Linitial + Lbug = Lfinal
The final moment of inertia of the system will be the sum of the initial moment of inertia of the disk and the bug's moment of inertia at the edge of the disk:
Ifinal = Idisk + Ibug
And the final angular velocity of the system can be calculated as:
wfinal = Lfinal / Ifinal
Now we can substitute the known values into the equations and calculate the new rate of spinning in rpm.