GATE CSE 2024 | Set 1 | Question: 44

Question

Consider a $512$ GB hard disk with $32$ storage surfaces. There are $4096$ sectors per track and each sector holds $1024$ bytes of data. The number of cylinders in the hard disk is _________.

Comments

No of track = no of cyclinder
32 Surface * x (let x track or cylinder ) * 4096 (sector per track) * 1024 (byte per sector) = 512GB
 

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Intuitive Solution with Visualisation.

 

Answer 1

Given that:

• Hard disk capacity= $512$GB=$2^9*2^{30}=2^{39}$ Byte
• Number of surface=$32=2^5$
• Sector/track=$4096=2^{12}$
• sector size/capacity=$1024=2^{10}$ Byte
• Number of cylinder =$x$

 we know that:

Disk capacity=Number of surface* number of track/surface* number of sector/track*number of byte/sector

$\implies2^{39}=2^5*x*2^{12}*2^{10}$
$\implies2^{39}=x*2^{12+10+5}$
$\implies2^{39}=x*2^{27}$
$\implies x=\frac{2^{39}}{2^{27}}=2^{12}=4096$

$\therefore$ the number of cylinders is $4096.$

Comments

> A cylinder is defined as the set of tracks that appear in the same location on each platter. 

i think they have asked for cylinders, not tracks. so wouldn't the final answer be 4096 / 32 ?

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@Shreyas_Jadhav

Disk capacity=Number of surface* number of track/surface* number of sector/track*number of byte/sector

here, $x = 4096$ means number of tracks per surface.

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Reference visual image ?

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@anujs

number of tracks per surface = #cylinders

Answer 2

We have -

capacity = #surfaces x #cylinders x #sectors x #capacity_of_sector

Putting all the values, we get -

$2^{39} = 2^5 * \text{ #cylinders } * 2^{12} * 2^{10}$

$\text{#cylinders} = 2^{39-27} = 2^{12} = 4096$

Answer - 4096

Answer 3

There are 32 storage surfaces meaning 16 plates with both sides capable of storing data. Now capacity of one plate is 512GB/32 = 16GB.

Now Cylinder is formed using tracks on a plate and there are sectors in each track and each sector stores data. So

Plate size = Total tracks * number of sectors per track * size of sector

16GB = Total tracks * 4096 * 1024

Total tracks or total cylinders = 4096 :)

Answer 4

Disk capacity = 512 GB

1 GB = 2³⁰ B

512 GB = 2³⁹ B

Sector size = 1024 B = 2¹⁰ B

Total number of sectors

= 2³⁹ / 2¹⁰

= 2²⁹ sectors

Given:

Number of sectors per track = 2¹²

Number of surfaces = 2⁵

Sector number calculation:

2¹² ) 2²⁹   ( 2¹⁷ 

      - 2²⁹

      --------

          0

Sector number = 0

Cylinder and surface calculation:

  2⁵ ) 2¹⁷ ( 2¹²

       - 2¹²

        -------

          0

Surface number = 0

Cylinder number = 2¹²⁼4096

Answer 5

ANS:4096

 

Answer 6

To simplify the calculation, we can express all the given parameters as powers of 2:

Total Capacity = $512 \text{ GB} = 512 \times 2^{30} \text{ bytes} = 2^9 \times 2^{30} \text{ bytes} = 2^{39} \text{ bytes}$

Storage Surfaces = $32 = 2^5$

Sectors per Track = $4096 = 2^{12}$

Bytes per Sector = $1024 = 2^{10} \text{ bytes}$

We can set up the mathematical equation using $C$ to represent the unknown number of cylinders in the hard disk:

$$\text{Total Capacity} = C \times \text{Surfaces} \times \text{Sectors per track} \times \text{Bytes per sector}$$

Substituting the powers of 2 values into this relationship gives:

$$2^{39} = C \times 2^5 \times 2^{12} \times 2^{10}$$

Combining the exponents on the right side simplifies the equation:

$$2^{39} = C \times 2^{27}$$

Solving for $C$ by dividing both sides isolates the cylinder parameter:

$$C = \frac{2^{39}}{2^{27}} = 2^{39 - 27} = 2^{12}$$

Converting the result back from a power of 2 :

$$C = 4096$$