Force and Pressure

Force

A force is any influence that can change the motion of a body. In particular, forces cause masses to accelerate, changing their velocity. Forces can be caused by a wide variety of phenomena, from the gravitational forces keeping planets in their orbits to the contact forces exerted on a door as we push or pull it. The force experienced by an object is defined as the rate of change of momentum of the object and is measured in units of Newton's \(\mathrm{N}\) which corresponds to \(\mathrm{kg}\,\mathrm{m}\,\mathrm{s}^{-2}\).

Forces are vector quantities, as they have both a magnitude and a direction corresponding to the strength of the force and the direction in which it acts. When multiple forces act on an object, we can find the resultant force acting on an object by summing the individual force vectors using vector addition. It's this resultant force that ultimately determines how the body's motion changes. Being vectors, we can represent forces in diagrams using labelled arrows showing the direction and magnitude of the force, these diagrams are often used when considering how multiple forces interact on one body.

Fig . 1 - Free-body diagram demonstrating the forces, weight and air resistance, acting on a falling box.

Consider two objects \(A\) and \(B\) interacting with each other, these interactions produce forces between the two bodies known as action and reaction forces. These action and reaction forces always have equal magnitudes and opposite directions. For example, when you push a door the force you exert on the door is equal to the force the door exerts on your hand, just acting in the opposite direction. This fundamental principle is known as Newton's third law.

We use these action-reaction forces all the time, for example when rowing a boat we exert a backward force on the water with the oars and the reaction force of the water propels the boat forwards. Similarly, jet engines fling high-velocity oxygen backward in order to propel a plane through the air.

It's important to remember, that just because two objects exert the same force on each other, it doesn't mean they experience the same acceleration. Objects with larger masses experience less acceleration, which is why the earth doesn't move downwards every time we take a jump!

Pressure

Pressure is a physical quantity closely related to force. Consider a force distributed over the surface area of an object, the magnitude of force applied perpendicularly to a unit surface area of the object defines the pressure exerted on the object. The smaller the surface area that a force acts upon, the greater the pressure will be. This is why a sharp knife, with a small blade surface area, can cut through something which a blunt knife could not even if the same force was applied to both knife.

Although force is a vector, pressure is a scalar expressing the magnitude of the perpendicular force on the unit surface area. Pressure is measured in units of Pascals \(\mathrm{Pa}\), equivalent to Newton's per square meter \(\mathrm{N}\,\mathrm{m}^{-2}\).

Pressure often arises in the context of fluids such as a gas within a container. Here, pressure is exerted upon the walls of the container by colliding gas molecules. Given Newton's Third Law, when a gas molecule collides with a container wall the wall exerts a reaction force onto the molecule causing it to accelerate back in the direction it came from. As such, the pressure exerted by a gas can be defined as the average rate of change of the gas particle's momentum as they collide with the walls of the container.

For example, the internal pressure exerted on a balloon by the gas molecules inside is what keeps it inflated. This pressure must be equal to the atmospheric pressure acting on the outside of the balloon for the balloon to remain stable. Any less than atmospheric pressure the balloon would implode, any more and the balloon would pop.

It's important to note that pressure exists within the entire gas system, not just at the walls of the container. For any static enclosed fluid, pressure is transmitted undiminished, meaning that changes in pressure in one region of the fluid cause equal changes in pressure everywhere in the fluid. This is known as Pascal's principle.

With these definitions of force and pressure in mind, let's look at how we can calculate force and pressure in different systems.

Force and Pressure: Formula

As we saw in the previous section, force is defined as the rate of change of momentum experienced by an object.

So if the momentum of an object changes by \(\Delta p\) in a time of \(\Delta t\) then the force on the object will be

\[F=\frac{\Delta p}{\Delta t}.\]

In the case where the mass of the object remains constant, as is usually the case, this simplifies to the famous expression associated with Newton's Second Law

\[F=ma\]

where \(m\) is the mass of the object and \(a\) is its acceleration. So, the acceleration experienced by an object under a force is dependent on the mass of the object.

From the definition of pressure, we see that the pressure and the force acting on a surface area are related by the following formula\[P=\frac{F}{A}.\]

This equation shows that at a fixed area, pressure and force are directly proportional, whereas the pressure associated with a fixed force is inversely proportional to the surface area the force is applied to.

We can use this definition of pressure to calculate pressures in a variety of contexts. For example, when you go swimming and dive deeper into the water, the water above you exerts a pressure on your body. By using the definition for pressure, we can calculate how water pressure changes with depth.

Consider a column of water stretching from the surface down to some depth \(h\), with a volume (V\). If the density of the liquid is \(\rho\), then the mass of the water within the column is\[m=\rho V.\]

The weight of this water is then

\[\text{Weight}=\rho g V.\]

with \(g=9.81\,\mathrm{m}\,\mathrm{s}^{-2}\) the acceleration due to gravity.

If the surface area at the bottom of the column is \(A\), then multiplying this area by the height of the column gives the volume\[V=Ah.\]

We know that the weight of the water in the column causes a pressure on the bottom of the column given by

\[P=\frac{\text{Weight}}{\text{Surface Area}}=\frac{\rho g V}{A}.\]

Cancelling the area term in the definition of the volume then gives

\[P=\frac{\rho g \cancel{A}h}{\cancel{A}}=\rho g h.\]

As we can see, water pressure increases linearly with depth, and this holds for any liquid.

Force and Pressure: Examples

Let's take a look at some of the interesting phenomena that can be explained with the concept of pressure.

Imploding Coke Cans

Whilst we never really notice the effects of standard atmospheric pressure, we can see its power with a simple experiment that you can do in a school lab. You'll need an empty fizzy drink can, a heat source like a Bunsen burner, some heat-proof tongs and a tub of ice-cold water. Pour a bit of tap water into the empty can and hold the can above the Bunsen burner. As the can heats up, the water will begin to boil and evaporate. After letting the water boil for a few seconds, using the heatproof tongs, take the can off the heat and quickly dunk it upside down in the ice-cold water. Incredibly, the coke can will crumple almost instantly!

To understand why this occurs, we need to consider the pressures acting on the can throughout the experiment. Initially, the pressure of the air outside the can is equal to the air pressure within the can, keeping it stable. As the water in the can boiled, it turned into vapor, pushing the air out of the can as it filled the can. Still, the pressure of the water vapor is equal to the air pressure outside the can. However, when the can is dunked in the cold water, the water quickly condenses back into a liquid, leaving a small vacuum inside the rest of the can with no gas inside. This low-pressure vacuum within the can means the air pressure outside the can is much greater than the pressure inside the can, causing the can to crumple under the air pressure.

Hydraulics

Hydraulic systems are systems that use liquids to produce mechanical power to do work. For example, construction machinery, such as diggers, use hydraulic motors and cylinders to move the digging arm.

Fig.4-Diggers use hydraulic motors and cylinders to produce forces capable of moving and lifting their crane arms.

Hydraulic systems use pascals principle of pressure transmission in liquids to amplify forces. To understand how this works, consider a simple hydraulic system composed of two plates with surface areas \(A_1\) and \(A_2\) connected by an enclosed tube of water, with \(A_1<A_2.\) If a force \(F_1\) is applied to the surface \(A_1\), the water pressure will increase by \[P=\frac{F_1}{A_1}.\] Pascals principle states that the water pressure will increase by \(P\) everywhere in the tube. This means that the pressure at the other plate \(A_2\) is also \(P\), and so the force pushing it up will be

\[F_2=PA_2.\] As \(A_2>A_1\) then the force \(F_2\) will also be larger, amplifying the initial force \(F_1\). This principle is used in hydraulic levers such as carjacks, where a small force applied to the lever manually is amplified so that it can lift a car up!

Similarities between Force and Pressure

Clearly, force and pressure are closely related concepts, so let's recap some key similarities between force and pressure.

• The force acting on a surface area and the pressure experienced by the object are directly proportional for a fixed surface area.

• If a system is at equilibrium, the forces acting on it must balance. Similarly, if an object such as a closed container remains stable, it means that the pressure inside the container is equal to the pressure outside the container.

• Force is equal to the rate of change of momentum experienced by a body. Similarly, the pressure exerted by a gas on the walls of a container is a measure of the average rate of change of the gas particles' momentum.

• Pressure can be calculated from the amount of force acting over a surface area \(A\) using the equation

  \[P=\frac{F}{A}.\]

Differences between Force and Pressure

However, it's important to remember that force and pressure are not equivalent quantities, as this can cause confusion when analyzing situations involving equivalent forces or pressures across different surface areas. Let's look at some key differences to keep in mind.