What must be the radius of a 4 kg circular disk if it is to have the same moment of inertia as a 1 kg rod 1 M long and pivoted at its midpoint?
To find the radius of the circular disk, we need to calculate the moment of inertia of both the disk and the rod, and then equate them.
The moment of inertia of a circular 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.
The moment of inertia of a rod pivoted at its midpoint is given by the formula: I = (1/12) * m * L^2, where I is the moment of inertia, m is the mass of the rod, and L is the length of the rod.
Let's denote the radius of the circular disk as R.
For the 4 kg circular disk:
I_disk = (1/2) * 4 kg * R^2 = 2 kg * R^2
For the 1 kg rod:
I_rod = (1/12) * 1 kg * (1 m)^2 = 1/12 kg * m^2
Now, we can equate the two expressions for the moment of inertia:
2 kg * R^2 = 1/12 kg * m^2
Since we want to find the radius of the disk, we need to isolate R. We know that the mass of the rod is 1 kg, so we can substitute that value into the equation:
2 kg * R^2 = 1/12 kg * (1 kg)^2
2 kg * R^2 = 1/12 kg
Now, divide both sides of the equation by 2 kg:
R^2 = 1/24 kg
To find the radius R, take the square root of both sides:
R = sqrt(1/24 kg)
Therefore, the radius of the circular disk would be sqrt(1/24) kg.