Question

The pressure on a surface is equal to the total force divided by the area. Write and simplify an expression for the pressure on a circular surface of area \(\pi d^{2} / 4\) subjected to a distributed load \(F\)

Short answer

The simplified expression for the pressure on a circular surface subjected to a load F is \(P = \frac{4F}{\pi d^2}\).

Step-by-step solution

Write the formula for pressure

The pressure (P) on a surface is calculated with the formula P = total force (F) divided by the area (A) on which the force is applied. The formula is expressed as P = F / A.

Substitute the given area into the pressure formula

For a circular surface with area \(A = \frac{\pi d^2}{4}\), substitute A in the pressure formula: P = F / \(A = F / \left(\frac{\pi d^2}{4}\right)\).

Simplify the expression

To simplify the expression, multiply both the numerator and denominator by 4 to get rid of the fraction in the denominator. This gives us P = \(\frac{4F}{\pi d^2}\).

Key concepts

Force Division

Understanding how to divide force correctly is essential when calculating pressure, which is a measure of force exerted per unit area. In physics, pressure is often the result of force distribution over a surface. For a given total force acting on a surface, force division is a straightforward calculation where the force is divided by the area to obtain the pressure.

Imagine having a book pressing down on a table. If we use the entire surface of the table to calculate the pressure, we’d get a very low value. However, if we only consider the contact area of the book, the pressure would be much higher. This highlights the significance of the area in the context of pressure. Thus, when we speak of force division in pressure calculation, we're not just splitting the force evenly, we're relating it to the exact area over which it's applied to find out how intensively the force acts upon that specific region.

Area of a Circle

The area of a circle is a key concept when dealing with circular surfaces in mathematics and physics. It's given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle. When the diameter \( d \) is used instead, the formula becomes \( A = \frac{\pi d^2}{4} \) because the diameter is twice the radius (\( d = 2r \) ).

In our exercise scenario, where a circular surface is subjected to a distributed load \( F \), knowing how to calculate the circle’s area allows us to pursue the pressure calculation. The reason the area is important is that it helps us understand the spread of force across the surface. The larger the area, the more spread out and thus, the lower the pressure for the same amount of force. When students wrap their head around this formula, they can approach not only pressure calculations but also numerous other practical problems involving circular areas with confidence.

Simplification of Expressions

Mastering the simplification of expressions is an invaluable skill in algebra that aids in understanding and solving various mathematical problems. Simplification is the process of altering a mathematical expression to make it easier to understand or work with, without changing its value. For instance, when an expression contains fractions, we often aim to remove the fraction by finding an equivalent expression that is simpler to deal with.

In the context of our exercise, simplifying the expression for pressure involves eliminating the fraction in the denominator. By multiplying both the numerator and the denominator of the pressure formula by the same number, we maintain the expression’s equality while achieving a more straightforward representation. It is crucial to remember that whatever operation is performed on the numerator must also be done on the denominator to keep the equation balanced. Ultimately, simplification allows us to arrive at an expression that is more manageable and easier to use in calculations or further analysis.