Inertia and Angular Acceleration

[CollegeStudent]

Homework Statement

Two concentric disks are welded together with a small hole through the center. This two disk unit is mounted onto a frictionless horizontal axle through the center hole. Two strings are attached and wound in opposite directions around the outer perimeter of each disk and are left hanging. The smaller disk has a mass of 4.77 kg and a diameter of 32.4 cm. The larger disk has a mass of 6.85 kg and a diameter of 44.2 cm. (a) If a mass of 400 g is hung from the string wrapped around the smaller disk, what mass should be hung from the string around the larger disk so that the two disk unit does not rotate? (b) If no mass is hung from the string around the larger disk, what will be the magnitude and direction of the acceleration of the two disk unit?

Homework Equations

The Attempt at a Solution

Not really sure what to do here...I can calculate the Inertia of the Smaller disk and the larger disk...But I don't know what to do.

I_disk = 1/2mR²

any hints here? Thanks in advance

 

[Yosty22]

The only thing I can think of is balancing the torques. If it doesn't rotate, it won't have any torque, so use the proper definition of torque to find when, with both masses and both moments, that the total torque on the system is 0.

 

[mfb]

For (a): Which property of the masses would "try" to rotate the disks? How can you get zero acceleration with those two masses?

In (b), just do the same consideration as in (a), but with a single mass and the total inertia of the system.

Edit: Yosty22 was quicker.

 

[tiny-tim]

Hi CollegeStudent!

(a) If a mass of 400 g is hung from the string wrapped around the smaller disk, what mass should be hung from the string around the larger disk so that the two disk unit does not rotate? (b) If no mass is hung from the string around the larger disk, what will be the magnitude and direction of the acceleration of the two disk unit?

For (a) you don't need moment of inertia at all, since nothing is moving …

just add the torques​

For (b), use τ_net = Iα

 

[Nickie]

.

To solve this problem, we need to use the concepts of inertia and angular acceleration. Inertia is the resistance of an object to changes in its state of motion, and angular acceleration is the rate of change of angular velocity. In this case, the two disks are connected by a frictionless axle, which means they can rotate freely.

(a) To find the mass that should be hung from the larger disk so that the two disk unit does not rotate, we need to consider the principle of torque. Torque is the rotational equivalent of force and is given by the formula:

τ = Iα

where τ is the torque, I is the moment of inertia, and α is the angular acceleration. In this case, the torque acting on the smaller disk will be equal and opposite to the torque acting on the larger disk, since they are connected by the same axle. Therefore, we can set up the following equation:

τ_small = τ_large

m_smallgr_small = m_largegr_large

where m_small and m_large are the masses hanging from the smaller and larger disks, respectively, g is the acceleration due to gravity, and r_small and r_large are the radii of the smaller and larger disks. We know the values for all these variables except for m_large, so we can rearrange the equation to solve for it:

m_large = (m_small * r_small * g) / (r_large * g)

Plugging in the values given in the problem, we get:

m_large = (0.4 kg * 0.162 m * 9.8 m/s²) / (0.221 m * 9.8 m/s²) = 0.6 kg

Therefore, a mass of 0.6 kg should be hung from the string wrapped around the larger disk to keep the two disk unit from rotating.

(b) If no mass is hung from the string around the larger disk, the torque acting on the smaller disk will be greater than the torque acting on the larger disk, causing the two disk unit to rotate in the direction of the smaller disk. The magnitude of the angular acceleration can be found using the same equation as before:

τ_small = τ_large

m_smallgr_small = m_largegr_large

α = (m_small * g) / I

Since no mass is hanging from the larger disk, m_large = 0, and the equation becomes:

α = (m_small * g