Density and fluid behaviour (Topic 4A)

The density of a substance is defined as its mass per unit volume.

Density relates the mass of a material to how much space it takes up.

Boxes A, B, and C all contain the same type of particle (with the same mass). Each box has the same volume but different masses, which are due to the different number of particles inside.

Box C has the greatest number of particles and therefore has the greatest mass. Since the boxes each have the same volume, Box C has the greatest mass per unit volume so it has the greatest density.

The density, of an object can be determined by measuring both its mass, and volume, with the equation:

To measure the density of a liquid:

• First measure the volume using a measuring cylinder.
• The mass can be measured by separately placing the empty measuring cylinder and the cylinder filled with liquid on a weighing scale, and then calculating the mass difference.

To measure the density of a regular solid:

• The volume can be calculated by using a ruler to measure the relevant dimensions.
• The mass of the solid can be measured on a weighing scale.

The density of an object can be determined by measuring both its mass and volume with the equation:

To measure the volume of an irregular solid:

• The volume can be measured by submerging the object in a liquid in a measuring cylinder.
• The volume of liquid displaced is equal to the volume of the solid. For example, the solid shown in the diagram below has a volume equal to
• The mass of the solid can be measured on a weighing scale.

Pressure is defined as the force acting perpendicularly (at a right angle) on a surface, divided by its area.

The equation for pressure is:

where:

• is pressure in pascals,
• is force in newtons,
• is area in metres squared,

Pressure has SI units of but is also commonly measured in pascals, where:

An example of this in practice is when you squeeze the ends of a pencil: you would apply approximately the same force to either end, but the pressures would be different.

Applying the same force to a smaller area produces a higher pressure:

• The sharp tip of a pencil has a small area, resulting in high pressure.
• The flat end has a larger area, resulting in lower pressure.

A fluid can be either a liquid or a gas.

Fluid pressure acts equally in every direction at a given depth.

The pressure acts perpendicularly (at right angles) to the walls of any container.

Since the holes are positioned at the same depth in the water bottle, the pressure at each hole is equal, resulting in equal water flow paths.

It is important to note that a fluid must be at rest (not flowing) for the pressure at a given point to act equally in all directions.

The diagram demonstrates how pressure applied to an enclosed fluid is transmitted in the absence of a gravitational field. In this model, the container walls are assumed to have uniform strength and thickness, ensuring an even response to internal pressure.

The force, applied by the piston generates a pressure, within the liquid. Water squirts out in all directions with an equal flow rate because the pressure is transmitted equally throughout the liquid and to the container walls.

It is useful to note that in the presence of gravity, the fluid exerts more pressure on the bottom of the container than on the top, as pressure increases with depth due to the gravitational force acting on it.

The diagram shows a case without gravity, so the pressure is the same throughout the liquid. Therefore, even though the holes are at different positions, the water jets follow the same path perpendicular to the wall, because the pressure at each hole is the same.

The equation for pressure in a liquid can be derived by considering the pressure due to the weight of a column of liquid of height, on a cross-sectional area,

The volume of the column is and is the density of the liquid, so its mass is:

The force exerted by a mass, is:

Therefore, the pressure exerted by the column of liquid is:

Note that the pressure in a liquid does not depend on the cross-sectional area of the container.

Upthrust is the upwards force felt by an object in a fluid.

Upthrust is caused by the pressure difference between the upper and lower surfaces of an object.

The diagram above shows a box submerged in water. Each face of the box experiences water pressure, which remains constant at each depth.

The forces on the vertical sides of the box are equal and opposite, so they cancel each other out.

However, the forces on the top and bottom of the box are different: water pressure increases with depth, so the forces on the bottom are greater.

The box’s weight also acts downwards. If the box’s weight is larger than the force imbalance between the top and bottom, the object sinks. Otherwise, it will float.

The upthrust equation can be derived by considering the forces acting on an object submerged in a fluid.

The diagram below shows a cylindrical object of cross-sectional area, and length, submerged vertically in a fluid of density,

The object feels a force:

on its top surface and a force:

on its bottom surface.

Pressure increases with depth, so is greater than The upthrust is equal to the force difference:

The equation for pressure in a fluid is:

so the upthrust is:

The volume, of the object is:

Therefore, the upthrust can be written as:

The quantity, is the mass of the fluid displaced, so the upthrust is equal to the weight of the fluid displaced (using the weight equation where is mass).

Archimedes’ principle states that an object placed in a fluid feels an upthrust equal to the weight of the fluid displaced. This principle applies to all shapes of objects.

Archimedes’ principle applies whether an object is partially or fully submerged, determining whether it will float.

A floating object does not fully submerge; instead, it sinks to the level where enough liquid is displaced so that the upthrust equals its weight

It is important to note that an object partially submerged in water feels an upthrust force both due to the weight of displaced water and the weight of displaced air. However, the weight of air displaced is much less than the weight of displaced water, so the total upthrust can be approximated as the upthrust due to the displaced water only.

When an object moves through a fluid (liquid or gas), such as the air or water, it experiences a drag force due to the fluid.

Drag is a frictional force that opposes the motion of an object moving through the fluid, slowing the object down. Therefore, drag forces always act in the opposite direction to the object’s motion.

The drag force converts some of the object’s kinetic energy into thermal energy within the fluid due to the work done against the resistance of the fluid.

For example, some objects entering the Earth’s atmosphere from space burn up due to the very large speeds at which they travel. The object experiences a very large drag force, which results in large amounts of thermal energy.

When objects move through the air, they experience a drag force known as air resistance.

For example, cars and aeroplanes typically have smooth and streamlined shapes to reduce the amount of air resistance they experience. This allows the vehicle to travel at higher speeds while also reducing the amount of fuel consumed.

When an object is in free fall through a fluid, its weight remains constant throughout the fall. However, the drag force increases as the object’s speed increases.

At the instant an object begins to fall, the drag force is zero, and the total force acting on the object is due to its weight. The object accelerates at a rate equal to the acceleration due to free fall.

The image above shows that as the object falls, the speed increases, and so does the magnitude of the opposing drag force. The resultant force (net force) on the object decreases, and the instantaneous acceleration of the object decreases to less than

Eventually, the object will achieve constant speed due to the force of the weight and the force of the drag becoming equal. This is known as terminal velocity.

To investigate the motion of an object falling through a fluid under the influence of a drag force, you can use a motion sensor connected to a data logger or laptop.

In the setup below, a thin string passed over a pulley attaches the falling object to a light polystyrene ball. As the object falls through a liquid cylinder, such as water or glycerol, it pulls the polystyrene ball upwards.

The motion of the polystyrene ball in the air is identical to that of the object falling through the fluid, allowing you to analyse velocity–time and acceleration–time graphs without directly measuring the object’s motion in the liquid.

Pointing the motion sensor directly at the falling object in the fluid is not practical due to several limitations.

• The liquid can distort or scatter the sensor’s signal, resulting in inaccurate measurements.
• Small objects may be difficult for the sensor to track reliably, especially in viscous fluids where turbulence or bubbles can interfere.

Monitoring the polystyrene ball in the air eliminates these issues, as the sensor operates more effectively in the air and provides clean, accurate data while still reflecting the object’s motion in the fluid.