| Concept / Situation | Key Formula / Approach | Important Points |
|---|---|---|
| Hydrostatic Pressure | \(P=\rho g d\) (SI) or \(P=\gamma d\) (US units) | Pressure increases linearly with depth. Remember: density of water \(\rho=1000\,\text{kg/m}^3\) and weight density \(\gamma=62.5\,\text{lb/ft}^3\). |
| Hydrostatic Force | \(F=\int_{a}^{b} \rho g \, d(x)\, w(x)\, dx\) | Divide the submerged surface into thin strips; multiply pressure by area of strip. Endpoints and variable strip width (via similar triangles or geometry) are key. |
| Moments & Center of Mass (Discrete) | \(\bar{x}=\frac{\sum m_i\,x_i}{\sum m_i},\quad \bar{y}=\frac{\sum m_i\,y_i}{\sum m_i}\) | Balance of moments: the center of mass is the weighted average of positions. |
| Centroid of a Lamina | \(\bar{x}=\frac{1}{A}\int_a^b x\, f(x)\, dx,\quad \bar{y}=\frac{1}{A}\int_a^b \frac{[f(x)]^2}{2}\, dx\) | For regions with uniform density, the centroid is independent of density; symmetry simplifies the computation. |
| Pappus’ Theorem | \(V=A\,(2\pi\bar{x})\) and analogously for surface area | Volume generated by rotation equals the area of the region times the distance traveled by its centroid. |
In Section 3 of Chapter 8, we apply integral calculus to problems in physics and engineering. Two primary applications are covered:
Hydrostatic Pressure and Force: How pressure in a fluid increases with depth and how that pressure results in a net force on submerged surfaces (e.g., dams, tanks, plates). This involves breaking a surface into small elements, computing the force on each element, and integrating.
Moments and Center of Mass: Finding the balance point (centroid) of objects or regions; essential for understanding equilibrium and rotational behavior. The concept extends from discrete mass distributions to continuous laminae.
These ideas build on previous integration techniques (finding areas/volumes) and pave the way for analyzing more complex systems in physics and engineering.
By the end of this section, you should be able to:
Explain the relationship between depth and pressure in a fluid and compute hydrostatic pressure using \(P=\rho gd\) (or \(P=\gamma d\)).
Set up and evaluate integrals to calculate the hydrostatic force on various submerged surfaces.
Determine the center of mass for a system of particles and for continuous regions (laminae) using moments.
Employ the symmetry principle to simplify centroid calculations.
Apply Pappus’ Theorem to relate the centroid of a region to the volume (or surface area) of a solid of revolution.
Before tackling this section, ensure you are comfortable with:
Basic integration techniques and Riemann sums.
Geometry of shapes: understanding areas of rectangles, trapezoids, circles, etc.
Fundamental physics concepts such as force, weight, and the idea of pressure.
Basic properties of functions and symmetry, especially in relation to centroids.
Concept: The pressure at a depth \(d\) is given by \(P = \rho g d\) (SI) or \(P = \gamma d\) (US units).
Procedure:
Identify a small horizontal strip on the submerged surface.
Write the area of the strip, using geometry (e.g., similar triangles, circle equations).
Determine the depth \(d\) at that strip.
Express the differential force as \(dF = P \cdot dA = \rho g\,d(x)\,A(x)\,dx\).
Sum up all the forces using an integral over the appropriate interval.
Tip: Always draw a diagram and label key dimensions.
Concept: The center of mass is where the total moment equals the moment of the summed mass acting at a single point.
Procedure for Discrete Systems:
Multiply each mass \(m_i\) by its position \(x_i\) (or \(y_i\)).
Sum these products to obtain the moment.
Divide by the total mass: \(\bar{x} = \frac{\sum m_i\,x_i}{\sum m_i}\) (similarly for \(\bar{y}\)).
Procedure for Continuous Regions (Lamina):
Express a small element of mass \(dm = \rho\,dA\), where \(dA\) is the area element.
Find the moment of this element about the chosen axis.
Integrate over the entire region and divide by the total mass (or area if density is constant).
Tip: Use symmetry to reduce work—for symmetric regions the centroid lies along the axis of symmetry.
Concept: The volume of a solid of revolution can be determined by multiplying the area \(A\) of the region by the distance traveled by its centroid \(d=2\pi\bar{x}\) when rotated about an axis.
Formula: \(V = A\,(2\pi\bar{x})\).
Tip: Ensure that the region is completely on one side of the axis of rotation.
Flashcard 1:
Front (Q): How is the pressure at a depth \(d\) in a fluid calculated?
Back (A): \(P = \rho g d\) (or
\(P=\gamma d\) in US units); pressure
increases linearly with depth.
Flashcard 2:
Front (Q): What is the differential expression for hydrostatic
force on a small strip?
Back (A): \(dF = P\,dA = \rho
g\,d\,w(x)\,dx\), where \(w(x)\)
is the width of the strip and \(d\) is
the depth.
Flashcard 3:
Front (Q): Write the formula for the center of mass of a system
of discrete masses along the \(x\)-axis.
Back (A): \(\bar{x} = \frac{\sum m_i
\, x_i}{\sum m_i}\).
Flashcard 4:
Front (Q): How do you find the centroid of a lamina bounded by
\(y=f(x)\) above the \(x\)-axis?
Back (A): \(\bar{x} =
\frac{1}{A}\int_a^b x\,f(x)\,dx\) and \(\bar{y} = \frac{1}{A}\int_a^b
\frac{[f(x)]^2}{2}\,dx\), where \(A=\int_a^b f(x)\,dx\).
Flashcard 5:
Front (Q): What does Pappus’ Theorem state about the volume of
a solid of revolution?
Back (A): The volume is the product of the area \(A\) of the region and the distance
travelled by its centroid: \(V= A\,(2\pi
\bar{x})\).
Flashcard 6:
Front (Q): Why is it important to divide a submerged surface
into small strips when computing hydrostatic force?
Back (A): Because pressure varies with depth, using small
strips allows us to approximate each strip with nearly constant pressure
and integrate accurately.
Flashcard 7:
Front (Q): True or False: In a uniformly dense region, the
centroid’s coordinates depend on the density.
Back (A): False. If density is constant, it cancels out when
computing the centroid.
Flashcard 8:
Front (Q): What role does symmetry play in determining the
centroid?
Back (A): If a region is symmetric about a line, the centroid
will lie on that line, reducing the calculations needed.
Goal and Thought Process: Find the force on a trapezoidal dam submerged in water, where pressure increases with depth. Divide the dam into horizontal strips with width determined by similar triangles.
Steps:
Setup: Choose a coordinate system with \(x=0\) at the water surface and \(x\) increasing downward.
Determine Width: For a strip at depth \(x^*\), use similar triangles (e.g., \(a=8-\frac{x^*}{2}\)) to find width \(w(x^*)=46-x^*\).
Pressure: Compute pressure \(P=\rho g x^*\) (with \(\rho=1000\,\text{kg/m}^3\) and \(g=9.8\,\text{m/s}^2\)).
Force Element: The differential force \(dF=1000\,g\,(46-x^*)\,x^*\,dx^*\).
Integrate: Sum the contributions: \[F=\int_{0}^{16} 1000\,g\,(46x-x^2)\,dx.\]
Recap: This example models breaking a complex shape into simple horizontal strips, applying pressure that changes linearly with depth, and integrating to obtain the total force.
Goal and Thought Process: Locate the centroid of the region between the line \(y=x\) and the parabola \(y=x^2\) in the first quadrant.
Steps:
Area: Compute area: \[A=\int_0^1 (x-x^2)\,dx=\frac{1}{6}.\]
Centroid’s \(x\)-Coordinate: \[\bar{x}=\frac{1}{A}\int_0^1 x(x-x^2)\,dx=\frac{1}{A}\left[\frac{1}{12}\right]=\frac{1/12}{1/6}=\frac{1}{2}.\]
Centroid’s \(y\)-Coordinate: \[\bar{y}=\frac{1}{A}\int_0^1 \frac{1}{2}(x^2-x^4)\,dx=\frac{1}{A}\left[\frac{1}{10}\right]=\frac{1/10}{1/6}=\frac{3}{5}\,\, \text{(or as computed from the detailed integration)}.\]
Recap: By setting up the integrals for moments and area, then dividing, we calculate the centroid; symmetry and careful integration are crucial.
Practice these types of problems to solidify your understanding:
A rectangular plate 3 ft wide and 8 ft high is submerged with its top 2 ft below the water surface. Set up the integral to find the hydrostatic force on the plate.
A trapezoidal plate with known dimensions is submerged vertically. Explain how to approximate its force using similar triangles to determine the strip width.
Find the centroid of the region in the first quadrant bounded by \(y=\cos x\), \(y=0\), and \(x=0\) (as in Example 5 of the text).
Determine the centroid of the region bounded by the line \(y=x\) and the parabola \(y=x^2\).
Use Pappus’ Theorem to compute the volume of a torus formed by rotating a circle of radius \(r\) about a line a distance \(R>r\) from its center.
Always draw a clear diagram and label dimensions.
Write down the given formulas before starting.
Check units carefully (SI vs. US).
Simplify only when necessary and verify your integration limits.
Remember, the key to mastering these problems is to break them down into manageable steps. Focus on identifying the small elements (either force strips or differential area elements), set up your integrals clearly, and use symmetry wherever possible. Even if you feel anxious, following a systematic process will help ensure you don’t miss any crucial steps. You’ve got the tools—now trust your process and do your best on exam day!