# Rotational Dynamics Problem

1. The problem statement, all variables and given/known data

A hard drive consists of a rigid circular platter (or disk) of radius 0.05 m and thickness 0.002 m. In normal operating conditions, the hard drive spins at 7200 rotations per minute. The read/write head is a light metallic element that floats just above the platter and both senses and imposes the magnetic structure that embody the digital data stored on the disk. The head moves radially in and out across the disk to write data in concentric circular tracks that fill the surface of the platter. When the head crashes, it scrapes against the surface of the platter, destroying the disk. During the crash the read/write head scrapes against the disk providing a frictional force. Assume that just before the head crash the disk is rotating at 7200 rpm. You can ignore any friction at the bearing. For this situation, the frictional force due to the read/write head is perpendicular to the radius vector, so that the torque has a magnitude of T=r*F(f) ……(torque = radius * force of friction). The moment of inertia of a uniform disk is I=(1/2)(M)(R²).
01a.
If the head crashes while it is writing on one of the inner tracks, 0.0053 m from the axis, then what is the torque it exerts on the 0.0406 kg disk if the force of friction from the read/write head is 0.401N?

01b.
What is the angular acceleration associated with that torque?

02.
How long will it take the disk to stop spinning once the crash begins?

03.
How many revolutions does the disk undergo as it spins to a stop after the crash?

04.
If the head crashes when it is writing on one of the outer tracks, 0.033 m from the axis, then how long does it take for the disk to slow to a stop after the crash?

If anyone could explain how to attack this problem and show work, that would be so helpful. Thank you soooo much.

2. Relevant equations
•I=(1/2)(M)(R²)
•τ=r*F(f)
•τ=Iα
The 4 equations that relate:
1. initial angular velocity (ωi)
2. final angular velocity (ωf)
3. angular acceleration (α)
4. Δθ (honestly don’t know what this stands for…. change in distance?)
time (t)
•ωf=ωi + αt
•Δθ=ωi + (1/2)αt²
•Δθ=1/2(ωi+ωf)t
•ωf²=ωi²+2α(Δθ)

3. The attempt at a solution

For part 1a, I tried using the equation τ=r*F(f) and plugged in 0.0053 m for r and 0.401 N for F(f), and got τ=0.0021253 Nm. But is that the right radius to use? Would you instead use the 0.05 m radius? It seems like a ridiculous amount to me.
For part 1b, I used the equation τ=Iα, plugging in 0.0021253 Nm for τ and 0.005075 kg·m² for I (because I=(1/2)(M)(R²), so I=(1/2)(0.0406kg)(0.05²), so I=0.005075 kg·m²), and got α=0.4188 rad/s2.

Am I on the right track with this problem? My numbers just don’t sound right to me and I don’t know if I’m using the correct radius at some points. Any help is appreciated, thanks so much.

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