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For classical, old-style, non solid state hard disks (HDD), I know that random accesses are much slower than sequential accesses since the physical tip needs to jump from one point of the disk to another. However, I was wondering if random accesses also wear out disks by physically damaging the tip that needs to move more around the disk.

In other words, if I need to do some calculations on an hard disk that involve a large amount of random accesses, should I rather look for another, less efficient solution to avoid them? I would prefer to avoid breaking an hard disk with a week worth of continuous random accesses.

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"Damage" may not be the correct term, though "wear" may be.

A random read will not produce any more wear than any other sort of read or write.

Because the head does not actually touch the drive platters, there is no damage to the platters from moving the head back and forth across them.

However, the drive motors must run and the head must be actuated, which are all mechanical movement and therefore subject to normal mechanical wear.

Different sorts of drives have different reliability statistics. This is one of the primary difference in price between the various tiers of drives. Cheaper drives tend to have cheaper components that will wear out more quickly. Server-class drives tend to have top-end components that can go through a dizzying number of movements before they will tend to wear.

You can check out the datasheets for your particular drive to get a better idea of the MTTF (Mean Time To Failure) and the number of IO cycles it is designed to handle to see if your particular projects might be better suited to a more reliable drive.

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