Question

Pressure is defined as force per unit area. Yet Torricelli found that the diameter of the barometer dish and the surface area of contact between the mercury in the tube and in the dish did not affect the height of mercury that was supported. Explain this seemingly inconsistent observation in view of the relationship between pressure and surface area.

Short answer

The height of the mercury column is determined by balancing atmospheric pressure with the pressure due to the mercury column's height, independent of contact area.

Step-by-step solution

Define Pressure

Pressure is defined as the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Mathematically, it is given by the formula: \[ P =\frac{F}{A} \] where \( P \) is pressure, \( F \) is force and \( A \) is area.

Torricelli's Experiment

In Torricelli's experiment, a barometer is used, which consists of a tube filled with mercury inverted in a dish of mercury. It was observed that the height of the mercury column remained constant irrespective of the diameter of the dish or the surface area of contact.

Pressure at the Interface

The key observation is that the pressure at the interface between the mercury in the tube and the mercury in the dish must be the same on both sides. This is because fluid in equilibrium has the same pressure at any horizontal level.

Atmospheric Pressure

The height of the mercury column is actually a measure of atmospheric pressure. The weight of the mercury column creates a pressure at the base of the column that balances the atmospheric pressure on the mercury surface in the dish. This condition can be represented as: \[ P_{atm} = \frac{F_{mg}}{A} \] where \( P_{atm} \) is atmospheric pressure, \( F_{mg} \) is the force due to the weight of the mercury column, and \( A \) is the cross-sectional area of the tube.

Independence of Surface Area

Even if the diameter of the dish or the surface area of contact between mercury changes, the height of the mercury column remains the same because the pressure exerted by the mercury column depends only on its height and the density of mercury, not on the area.

Key concepts

Torricelli's experiment

Torricelli's experiment was a groundbreaking experiment in understanding pressure in fluids. By using a barometer, Torricelli filled a long glass tube with mercury and then inverted it into a dish also containing mercury. This setup led to the creation of a vacuum at the top of the tube, with the mercury column standing tall. What's fascinating is that the height of the mercury column remained the same no matter the size of the dish or the area of contact between the mercury in the tube and the dish.
This occurs because fluid pressure at a given depth in a fluid is the same throughout. Therefore, the surface area of the dish does not impact the height of the mercury column in the tube. This key observation explains an intrinsic property of pressure in fluids.
Hence, the height of the mercury column in Torricelli's experiment is not affected by the surface area of contact, demonstrating that fluid pressure at a horizontal level is uniform.

Atmospheric pressure

The atmospheric pressure plays a pivotal role in the equilibrium of fluids. Atmospheric pressure is the force exerted by the weight of the air above a surface. It acts equally in all directions and can support the mercury column in Torricelli's barometer.
In the experiment, atmospheric pressure pushes down on the surface of the mercury in the dish. This pressure is balanced by the weight of the mercury column inside the tube.
If we represent this condition, we get: This balance means that the atmospheric pressure at the surface of the dish ensures that the mercury column stays at a height where the weight of the mercury column, which depends only on the height and density of the mercury, equates to the external atmospheric pressure.
This relationship is given by \[ P_{atm} = \frac{F_{mg}}{A} \] where P_{atm} is atmospheric pressure, F_{mg} is the force due to the weight of the mercury column, and A is the cross-sectional area of the tube.

Fluid equilibrium

Fluid equilibrium occurs when the fluid is at rest, and there is no net flow of the fluid. In the context of Torricelli's experiment, the equilibrium is attained when the pressure within the mercury column equals the atmospheric pressure acting on the surface of the mercury in the dish.
The concept of fluid equilibrium can be understood by recognizing that at any horizontal level within a fluid, the pressure must be constant. Hence, a taller column of mercury standing still, without overflowing or retracting, indicates that the forces due to fluid pressure and external pressure are balanced.
This concept directly relates to why the diameter of the dish or the surface area of contact does not affect the height of the mercury column. The fluid pressure is determined only by the height and density of the fluid column and is independent of the surface area.
The principle of fluid equilibrium, illustrated by Torricelli's experiment, helps us understand various natural and man-made systems where fluid pressure needs to be balanced, such as in designing dams, hydraulic systems, and even understanding weather patterns.