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In this Physics tutorial, you will learn:

- The meaning of pressure
- Factors affecting the pressure
- The unit of pressure and its representation in SI units
- Why pressure is a scalar?

Doctors often prescribe intramuscular injections when people get sick. Why nurses use thin needles when they jab a patient?

Why thin drill bits are more suitable than thick ones to open holes in the wall?

Why workers often put the concrete mixer on a wide plate (caterpillar track) instead of putting it directly on the ground?

I'm sure you can give convincing answers in your words to all the above questions. However, you will be even more convincing if you are able to explain the abovementioned situation in scientific terms. Therefore, if you are interested in such things, please read the following paragraphs in which the pressure-related phenomena are discussed.

Obviously, there is a common explanation in simple words for all the above questions: A thin solid object penetrates much easier in another solid than a thicker one. This means a smaller force is needed to make a thin object punch a surface compared to the force needed for the same action if we used a thick object. On the other hand, if we have two identical pins and we are trying to nail them on a hard surface, the pin on which we use a greater force will penetrate easier inside the surface. This means the penetrating ability of an object varies directly with the force used and indirectly with its thickness.

The physical quantity that takes into consideration both the abovementioned factors is known as **"pressure, P"**. By definition, **"pressure represents the perpendicular force exerted in the unit of area"**.

Mathematically, we have

P = *F*_{⊥}*/**A*

Look at the figure.

If the force is not perpendicular to the surface, we consider only the normal component to it. This means we must also consider the sine of the angle formed by the force vector and the surface. Look at the figure.

Based on the figure, we can write

P = *F × sin θ**/**A*

In many cases, the force is already normal to the surface. Therefore, the angle is not written in the formula, as sin 900 = 1.

The unit of pressure is [N / m^{2}] as force is measured in Newtons and area in square metres. This unit is otherwise known as Pascals [Pa].

Let's resolve Pascal into SI units only. Thus,

1 [Pa] = 1[*N**/**m*^{2}] = 1 [*kg × m/s*^{2}*/**m*^{2}] = 1[*kg**/**m × s*^{2}]

What is the minimum force one must use (only by pushing, not hitting) to drive a nail into a solid surface if this surface can hold a maximum pressure of 4 000 000 Pa? The nail tip is 2 mm^{2} thick.

The minimum force is applied when it is exerted normal to the surface, i.e. θ = 90 and therefore, sin θ = 1.

Let's write both pressure and area in standard form. Thus,

P_{max} = 4 000 000 Pa = 4 × 106 Pa

A = 2 mm^{2} = 2 × 106 m^{2}

F = ?

We have:

P = *F**/**A* ⇒ F = P × A

= 4 × 10^{6} Pa × 2 × 10^{-6} m^{2}

= 8 N

= 4 × 10

= 8 N

Let's see how the surface of contact affects the value of solid pressure.

Compare the pressures exerted by the same object (a 12 kg cuboid of dimensions 4 dm × 3 dm × 2 dm). For convenience, take g = 10 m/s^{2}.

Since the opposite faces of a cuboid are identical, there are three possible positions it can be placed on the ground. They are:

For the area A, in all three cases we must consider only the lower base, as only it relies on the ground. Thus, In the first position, we have

A_{1} = 4 dm × 2 dm

= 8 dm^{2}

= 0.08 m^{2}

= 8 dm

= 0.08 m

In the second position, we have

A_{2} = 4 dm × 3 dm

= 12 dm^{2}

= 0.12 m^{2}

= 12 dm

= 0.12 m

and in the third position, we have

A_{3} = 3 dm × 2 dm

= 6 dm^{2}

= 0.06 m^{2}

= 6 dm

= 0.06 m

Force is the same in all cases; it is equal to the object's weight. Therefore, we have

F_{1} = F_{2} = F_{3}

= F

= W

= m × g

= 12 kg × 10 m/s^{2}

= 120 N

= F

= W

= m × g

= 12 kg × 10 m/s

= 120 N

Hence, we obtain for the pressure in all three positions:

P^{1} = *F*_{1}*/**A*_{1}

=*120 N**/**0.08 m*^{2}

= 1500 Pa

P^{2} = *F*_{2}*/**A*_{2}

=*120 N**/**0.12 m*^{2}

= 1000 Pa

P_{3} = *F*_{3}*/**A*_{3}

=*120 N**/**0.06 m*^{2}

= 2000 Pa

=

= 1500 Pa

P

=

= 1000 Pa

P

=

= 2000 Pa

Thus, again it is confirmed the fact that the smallest contact area produces the greatest pressure for the same force and vive-versa, i.e. force and area are inversely proportional to each other.

**Remarks! **

**1** There is a confusion derived from the fact that despite force is a vector, pressure is a scalar quantity. There are two explanations for this issue: a mathematical and a physical explanation. Let's discuss both of them.

**1 a** - In vector theory, area is represented by a vector that is normal to it, and this vector has a magnitude equal to the numerical value of the given area. Thus, if the area in the figure below is 7 cm^{2}, it is represented by a vector which is 7 units long and lies normal to the surface as shown in the figure below.

As a result, we have to divide two vector quantities when calculating pressure. It is known from section 2 that vectors can divide only in scalar mode. Therefore, when dividing two vectors such as force and area, we obtain a scalar, i.e. pressure

.**1 b** - We will explain in the next tutorials that - except for solids - pressure is a quantity, which acts in all directions. Thus, when we immerse in water, it exerts pressure in all directions on our body. This is also true for the gas pressure as well. For example, atmosphere - which is a mixture of gases - exerts the same pressure on our body in all directions. Therefore, it is not fair to consider pressure as a vector only because of solid pressure. Hence, definitively pressure is a scalar.

**2** - It is still unclear for most people whether pressure should be written in uppercase "P" or lowercase "p" in the formulae. There is a widely accepted opinion that one can use both versions depending on the topic that is being discussed. Thus, pressure in thermodynamics can be written in either capital 'P' or lowercase 'p' depending on how it's used in different engineering problems. However, we can also express pressure in capital 'P' when dealing with Enthalpy or Entropy.

By definition, **"pressure represents the perpendicular force exerted in the unit of area"**.

The smallest contact area produces the greatest pressure for the same force and vive-versa, i.e. force and area are inversely proportional to each other.

Mathematically, we have

P = *F*_{⊥}*/**A*

If the force is not perpendicular to the surface, we consider only the normal component to it. Thus, the equation of pressure becomes

P = *F × sin θ**/**A*

where θ is the angle formed by the force and the surface direction.

In many cases, the force is already normal to the surface. Therefore, the angle is not written in the formula, as sin 900 = 1.

The unit of pressure is [N / m^{2}] as force is measured in Newtons and area in square metres. This unit is otherwise known as Pascals [Pa].

In the SI system of units,

1 [Pa] = 1[*kg**/**m × s*^{2}*]*

Despite the force is a vector, pressure is a scalar because it acts in all directions. Another factor why pressure is consider as a scalar is the vector meaning sometimes we give to the surface by expressing it through a vector normal to it. Therefore, we have to divide in scalar mode two collinear vectors; force and area whose result gives a scalar, i.e. the pressure.

*1. A 4 kg cube of side length 2 dm rests on a horizontal 1 m × 1m square plate, as shown in the figure.*

What is the pressure exerted on the ground if the plate is massless? Take g = 10 m/s^{2}.

- 0.5 Pa
- 5.0 Pa
- 40.0 Pa
- 1000.0 Pa

**Correct Answer: C**

*2. A 70 kg skier is standing on his skis as shown in the figure.*

If the dimension of each ski is 1.4 m × 15 cm, calculate the pressure exerted by this man on the snow. For convenience, take g = 10 m/s^{2}.

- 333.3 Pa
- 3333.3 Pa
- 166.7 Pa
- 1666.7 Pa

**Correct Answer: D**

*3. A 40 N force is required to drive the tip of a nail in the wall. The nail's tip is 0.5 mm ^{2} thick. What is the force to be used when trying to drive the same nail from the other side, i.e. to drive its head in the wall? The head is 8 mm^{2} thick.*

- 640 N
- 160 N
- 10 N
- 2.5 N

**Correct Answer: A**

We hope you found this Physics tutorial "Pressure. Solid Pressure" useful. If you thought the guide useful, it would be great if you could spare the time to rate this tutorial and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of physics and other disciplines. In our next tutorial, we expand your insight and knowledge of Density And Pressure with our Physics tutorial on Liquid Pressure. Pascal's Principle.

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