Question

Suppose you pump the air out of a paint can, which is covered by a lid. The can is cylindrical, with a height of \(22.4 \mathrm{~cm}\) and a diameter of \(16.0 \mathrm{~cm}\). How much force does the atmosphere exert on the lid of the evacuated paint can? a) \(9.81 \mathrm{~N}\) c) \(2030 \mathrm{~N}\) b) \(511 \mathrm{~N}\) d) \(8120 \mathrm{~N}\)

Short answer

Answer: (c) 2030 N

Step-by-step solution

Find the Area of the Lid

To find the area of the lid, we need to use the area of a circle formula, which is given by \(A = \pi r^2\) where \(r\) is the radius of the circle. Since the diameter of the lid is \(16.0 \mathrm{~cm}\), we have the radius as half of it, which is \(r = 8.0 \mathrm{~cm}\). Now we can find the area of the lid: \(A = \pi (8.0 \mathrm{~cm})^2\)

Convert the Area to Square Meters

Since we will be using atmospheric pressure in Pascals, which are expressed in terms of Newtons per square meter, we should convert the area we calculated in square centimeters to square meters. There are \(10^4 \mathrm{~cm}^2\) in a square meter, so we can make the conversion. \(A = (\pi (8.0 \mathrm{~cm})^2)\times \frac{1 \mathrm{~m}^2}{10^4 \mathrm{~cm}^2}\)

Calculate the Force on the Lid

Now that we have the area of the lid in square meters, we can calculate the force exerted by the atmosphere on the lid. The force is the product of the area and the atmospheric pressure (\(F = P \times A\)). The atmospheric pressure on Earth at sea level is \(101,325 \mathrm{~Pa}\). So, we can plug in our numbers to calculate the force: \(F = 101,325 \mathrm{~Pa} \times (\pi (8.0 \mathrm{~cm})^2)\times \frac{1 \mathrm{~m}^2}{10^4 \mathrm{~cm}^2}\)

Evaluate the Force on the Lid

Now we can plug in the values into our calculator to find the force exerted: \(F ≈ 2030.28 \mathrm{~N}\) Now, rounding to the closest answer choice given in the exercise: \(F ≈ 2030 \mathrm{~N}\) Therefore, the correct answer is (c) \(2030 \mathrm{~N}\).

Key concepts

Force Calculation

Understanding the calculation of force is essential in physics, and it plays a pivotal role when we are dealing with atmospheric pressure. Force can be understood as a push or a pull upon an object resulting from its interaction with another object. When we are dealing with atmospheric pressure, we calculate the force exerted by the air above us.

As demonstrated in the original problem, to find the force that the atmosphere exerts on an object, such as the lid of a paint can, we use the formula: \( F = P \times A \), where \( F \) is the force in Newtons, \( P \) is the atmospheric pressure in Pascals, and \( A \) is the area over which the pressure is applied in square meters. This is the basic computation that allows us to understand how much force is being exerted by the atmosphere on a given surface.

Area of a Circle

Calculating the area of a circle is a common task in geometry. The formula to calculate the area is: \( A = \pi r^2 \), where \( A \) is the area, \pi \) is a mathematical constant approximately equal to 3.14159, and \( r \) is the radius of the circle. It is important to note that the radius is half of the diameter of the circle, which is a common oversight in many calculations.

For example, if you have a cylindrical object such as a paint can lid with a given diameter, you would first need to divide the diameter by two to find the radius, then plug that value into the area formula. In practical scenarios where you need the area in different units, the calculated area in square centimeters can be easily converted to square meters since there are \( 10^4 \) square centimeters in a square meter.

Pascals

Pascals (Pa) is the SI unit of pressure and is equivalent to one Newton per square meter. It is named after the French mathematician and physicist Blaise Pascal. This unit of measurement is commonly used in science and engineering to quantify internal pressure, stress, Young's modulus, and tensile strength.

For simplicity, atmospheric pressure at sea level is often represented as \( 101,325 \) Pascals, which is also equivalent to 1 atmosphere (atm). This standardized measure allows us to calculate the force exerted by the atmospheric pressure when multiplied by a specific surface area, as is necessary in many physics and engineering calculations.

Pressure and Force Relationship

The relationship between pressure and force is straightforward: pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Expressed mathematically, \( P = \frac{F}{A} \), where \( P \) is pressure, \( F \) is force, and \( A \) is the area.

This relationship explains why a sharp knife cuts better than a dull one: even with the same amount of force, the pressure exerted by the sharp knife is higher because it is applied over a smaller area. In the context of atmospheric pressure, this relationship allows us to calculate the total force exerted on an object's surface, such as a paint can lid, which illustrates how vast the air's weight is above us and how it applies a significant force even over small areas.