Four items, all with equal masses, are used to construct the device depicted to the right. Two of the items are uniform solid disks, the smaller of which has half the radius of the other. The disks are glued together so that they share a common axle. The other two items are blocks, which are attached to (massless) ropes, each wound around a different disk, as shown in the diagram. The whole system begins at rest,with the block attached to the smaller disk starting at a lower elevation than the other block. The blocks are released and begin to fall, while the disks turn and the ropes unwind (the disks are glued together, so they turn at equal speeds at all times). Find the rotational speed of the disks (in rad/s) when the blocks reach the same height.The radius of the smaller disk is R = 0.60m, and the blocks start at a height difference

Question

Four items, all with equal masses, are used to construct the device depicted to the right. Two of the items are uniform solid disks, the smaller of which has half the radius of the other. The disks are glued together so that they share a common axle. The other two items are blocks, which are attached to (massless) ropes, each wound around a different disk, as shown in the diagram. The whole system begins at rest,with the block attached to the smaller disk starting at a lower elevation than the other block. The blocks are released and begin to fall, while the disks turn and the ropes unwind (the disks are glued together, so they turn at equal speeds at all times). Find the rotational speed of the disks (in rad/s) when the blocks reach the same height.The radius of the smaller disk is R = 0.60m, and the blocks start at a height difference

of d = 1.5m.

Answer 1

To find the rotational speed of the disks when the blocks reach the same height, we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system remains constant if no external forces are acting on it.

Let's consider the initial and final states of the system.

Initial state:
The blocks are at a height difference of d = 1.5m. The total initial mechanical energy is given by the potential energy of the blocks:
E_initial = mgh, where m is the mass of each block, g is the acceleration due to gravity, and h is the height difference.

Final state:
When the blocks reach the same height, their potential energy will be zero. At this point, all the initial potential energy has been converted to the rotational kinetic energy of the disks. The total final mechanical energy is given by the rotational kinetic energy of the disks:
E_final = (1/2)Iω^2, where I is the moment of inertia of each disk and ω is the angular speed of the disks.

Since the disks are glued together, they rotate at the same speed, so ω is the same for both disks.

Now, we can equate the initial and final mechanical energies:

mgh = (1/2)Iω^2

Now, let's solve for ω:

First, we need to express I in terms of the mass and radius of the disks. The moment of inertia of a solid disk is given by:
I = (1/2)mr^2, where m is the mass of the disk and r is its radius.

Since the smaller disk has half the radius of the larger disk, we can denote the mass of each disk as m and the radii as r (larger disk radius) and r/2 (smaller disk radius).

Substituting the values into the equation:

mgh = (1/2)(1/2)mr^2ω^2
2mgh = (1/4)mr^2ω^2
8gh = r^2ω^2

Now, solve for ω:

ω^2 = (8gh) / r^2
ω = √((8gh) / r^2)

Substituting the given values: g = 9.8 m/s^2, h = 1.5 m, and r = 0.6 m:

ω = √((8 * 9.8 * 1.5) / (0.6)^2)

Simplifying the expression:

ω = √(117.6 / 0.36)
ω = √326.67
ω ≈ 18.07 rad/s

Therefore, the rotational speed of the disks when the blocks reach the same height is approximately 18.07 rad/s.