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?
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:
The rim accelerates faster than the disk in response to the torque.
Explanation:
Given:
For the rim:
mass = M
radius = R
The moment of inertia is:
[tex]I_{A} =MR^{2}[/tex]
For the disc:
mass = M
radius = 2R
The moment of inertia is:
[tex]I_{B} =\frac{1}{2} M(2R)^{2} =2MR^{2}[/tex]
If the same torque is applied, thus:
[tex]\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}[/tex]
According to this result, the rim accelerates faster than the disk.
Answer:
The second wheel
Explanation:
The torque is given by
[tex]\tau=I\alpha[/tex] (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:
[tex]I_r=mR^2\\I_d=\frac{1}{2}mR^2[/tex]
where we used that both wheel have the same mass. By replacing in (1) we obtain:
[tex]\alpha_r=\frac{\tau}{I_r}=\frac{\tau}{mR^2}\\\alpha_d=\frac{\tau}{I_d}=\frac{\tau}{\frac{1}{2}mR^2}=2\alpha_r\\[/tex]
Hence, the second wheel (the disk) has a greater acceleration.
hope this help!!