Question

g A has all the mass at the rim, while wheel B has the mass uniformly distributed, like a solid disk. The wheels have the same mass. The same torque is applied to both wheels. Which one accelerates faster in response to this torque?

Answer 1

The rim accelerates faster than the disk in response to the torque.

Step-by-step explanation:

Given:

For the rim:

mass = M

radius = R

The moment of inertia is:

`I_(A) =MR^(2)`

For the disc:

mass = M

radius = 2R

The moment of inertia is:

`I_(B) =(1)/(2) M(2R)^(2) =2MR^(2)`

If the same torque is applied, thus:

`\tau _(A) =\tau _(B) \\I_(A)\alpha _(A) =I_(B)\alpha _(B) \\MR^(2)\alpha _(A)=2MR^(2)\alpha _(B) \\\alpha _(A)=2\alpha _(B)`

According to this result, the rim accelerates faster than the disk.

Answer 2

The second wheel

Step-by-step explanation:

The torque is given by

`\tau=I\alpha` (1)

where I is the moment of inertia and a is the angular acceleration. If we take into account the moment of inertia of a disk and a ring ()for the first wheel) we have:

`I_r=mR^2\\I_d=(1)/(2)mR^2`

where we used that both wheel have the same mass. By replacing in (1) we obtain:

`\alpha_r=(\tau)/(I_r)=(\tau)/(mR^2)\\\alpha_d=(\tau)/(I_d)=(\tau)/((1)/(2)mR^2)=2\alpha_r\\`

Hence, the second wheel (the disk) has a greater acceleration.

hope this help!!