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The document provides instructions for solving related rates problems involving a spherical balloon deflating. It identifies finding the rate air is removed when the radius is 9 cm as the problem. It sets up the equation relating volume, radius, and their rates of change and takes the derivative. Substituting the known radius and radius rate allows solving for the required air removal rate of -4860π cm3/min.
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A spherical balloon is deflating, with its radius decreasing at a constant rate of 15cm/min. The document calculates that when the balloon's radius is 9cm, the rate at which air must be removed is 4860π cm^3/min in order for the radius to decrease at the given rate. It uses the formula for the volume of a sphere and takes the derivative of volume with respect to time to determine the required rate of air removal.
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Green's theorem relates a line integral around the boundary of a plane region D to a double integral over D. Specifically, if P and Q have continuous partial derivatives on an open region containing D, and C is the positively oriented boundary of D, then the line integral of P dx + Q dy around C equals the double integral over D of (∂Q/∂x - ∂P/∂y) dx dy. Green's theorem can be extended to any region that can be decomposed into piecewise smooth, simple closed curves. The flux form of Green's theorem relates the line integral around C of a vector field to the double integral of the curl of the vector field over D.
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The document discusses key concepts in trigonometry including the unit circle, radians, converting between degrees and radians, trigonometric functions and their domains and ranges, trigonometric identities, graphing trigonometric functions, and inverses of trigonometric functions. It provides examples to illustrate how to use radians to find arc length and linear/angular speed, evaluate trig functions at specific angles, use trig identities and Pythagorean identity, and take inverses of trig functions with domain restrictions.
Circumference and area of cirlce
This document defines perimeter, circumference, and area of different shapes. It provides the following definitions:
Perimeter is the distance around a geometric figure. Circumference is the distance around a circle, which is calculated by multiplying the diameter by pi. The area of a rectangle is calculated by multiplying its length by its width. The area of a triangle is half the product of its base and height. The area of a circle is calculated by multiplying pi by the square of the radius. Examples are given for calculating each measurement.
Perimeter area circumference
This document defines perimeter, circumference, and area of different shapes. It provides the following formulas:
Perimeter is the distance around a figure and is calculated by adding all the side lengths. Circumference is the distance around a circle and is calculated using the formula C=πd, where d is the diameter.
The area of a rectangle is calculated by multiplying its length and width (A=l×w). The area of a triangle is half the product of its base and height (A=1/2bh). The area of a circle is the product of π and the square of the radius (A=πr^2).
Differential calculus
1) The document discusses concepts in differential calculus including the product rule, quotient rule, derivatives of trigonometric functions, higher-order derivatives, and polar curves.
2) Polar curves are represented using polar coordinates (r, θ) where r is the distance from the origin and θ is the angle. Conversions between polar and Cartesian coordinates are covered.
3) Methods for calculating the area inside a polar graph, length of a polar curve, and surface area obtained by rotating a polar curve about the x-axis are presented using integral calculus.
Trying to rearrange the VdW EoS exactly to get V .pdf
Trying to rearrange the VdW EoS exactly to get V as a function of T and P so you
can then differentiate it explicitly results in a mess (you need to solve a 3rd order polynomial in
V). An easier method is to find dT/dV)p, and then take the inverse. From the EoS, we have: T =
(1/R)*(P+a/V^2)*(V-b) dT/dV)p = (P*V^3 - a*V +2*a*b)/(R*V^3) dV/dT)p =
(R*V^3)/(P*V^3 - a*V +2*a*b) This is an exact result. Unfortunately, it\'s probably not the
result your professor is looking for. I\'ll derive an approximate result below, which is what one
needs to prove the second part of this question. -------------- The J-T coefficient is defined as
dT/dP)H (i.e., dT/dP at constant enthalpy). From the basic relationships among partial
derivatives: dT/dP)H = -dH/dP)T / dH/dT)p But we know from its definition that for a 1-
component system: dH = TdS + VdP so dT/dP)H = [ -T*dS/dP)T - V*dP/dP)T]/[T*dS/dT)p +
V*dP/dT)p] = [-T*dS/dP)T - V]/[T*dS/dT)p] From one of the Maxwell relations, we know that
dS/dP)T = -dV/dT)p, and we also know that Cp = T*dS/dT)p, so: dT/dP)H = [T*dV/dT)p -
V]/Cp We now need to evalutate this expression for a VdW gas. The expression given in the
question for the J-T coefficient for the VdW gas is *not* an exact result, which you can easily
show by plugging the expression for dV/dT)p found above into the equation for dT/dP)H. In
order to get the desired result, we need back up and make some approximations in the derivation
of dV/dT)p Going back to the EoS, we can rewrite it as: P*V = R*T - a/V + b/P + a*b/V^2 If
we ignore the small second order terms in the VdW coefficients (i.e., a*b/V^2), then V ~=
R*T/P - a/(P*V) + b If we now make another appproximation and assume P*V ~= R*T (i.e.,
assume the gas behaves close to ideally), we can write: V ~= R*T/P - a/(R*T) + b Taking the
derivative with respect to T at constant P: dV/dT)p ~= R/P + a/(R*T^2) Plugging this new,
approximate expression for dV/dT)p into the J-T coefficient equation, we have: dT/dP)H =
[T*dV/dT)p - V]/Cp ~= [R*T/P + a/(R*T) - R*T/P + a/(R*T) - b]/Cp dT/dP)H = ~= [2*a/(R*T)
- b]/Cp which is the desired result.
Solution
Trying to rearrange the VdW EoS exactly to get V as a function of T and P so you
can then differentiate it explicitly results in a mess (you need to solve a 3rd order polynomial in
V). An easier method is to find dT/dV)p, and then take the inverse. From the EoS, we have: T =
(1/R)*(P+a/V^2)*(V-b) dT/dV)p = (P*V^3 - a*V +2*a*b)/(R*V^3) dV/dT)p =
(R*V^3)/(P*V^3 - a*V +2*a*b) This is an exact result. Unfortunately, it\'s probably not the
result your professor is looking for. I\'ll derive an approximate result below, which is what one
needs to prove the second part of this question. -------------- The J-T coefficient is defined as
dT/dP)H (i.e., dT/dP at constant enthalpy). From the basic relationships among partial
derivatives: dT/dP)H = -dH/dP)T / dH/dT)p But we know from its definition that for a 1-
component system: dH = TdS + VdP so dT/dP)H = [ -T*dS/dP)T - V*dP/dP)T.
EES Functions and Procedures for Forced convection heat transfer
This file contains notes on Engineering Equation Solver (EES) Functions and Procedures for Forced convection heat transfer calculations. Some problems are also included.
These notes were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
It is hoped that these notes will be useful to teachers, students, researchers and professionals working in this field.
Contents:
• Forced convection – Tables of formulas
• Boundary layer, flow over flat plates, across cylinders, spheres and tube banks –
• Flow inside tubes and ducts
Integration application (Aplikasi Integral)
Materi kuliah tentang Aplikasi Integral. Cari lebih banyak mata kuliah Semester 1 di: http://muhammadhabibielecture.blogspot.com/2014/12/kuliah-semester-1-thp-ftp-ub.html
23 Double Integral over Polar Coordinate.pptx
The document discusses double integrals over polar equations and cylindrical coordinates. It explains that a double integral over a region D defined in polar coordinates by r1(θ) and r2(θ), where r1 < r2 and A < θ < B, can be evaluated as an integral of an integrand function f(r,θ) over the region D in terms of drdθ. It also introduces the cylindrical coordinate system, which uses polar coordinates for points in the xy-plane and z as the third coordinate, and provides examples of converting between rectangular and cylindrical coordinates and sketching curves defined by constant r and θ values.
Chapter 03 drill_solution
1. The document summarizes solutions to problems from chapter 3 of Engineering Electromagnetics by Hayt, Buck. It provides calculations and solutions for problems regarding electric field intensity D, electric flux, volume charge density, and more.
2. Key calculations include determining D at a point given point charges, determining electric flux through surfaces, calculating enclosed charge, and finding volume charge density using the divergence of D.
3. Solutions involve applying Gauss's law and knowing how to set up integrals in different coordinate systems to calculate relevant physical quantities for the electromagnetic problems.
Rates of change_updated
The document discusses the concept of derivative as a rate of change and provides examples of how it can be used to analyze quantities that change with respect to other variables, such as position, velocity, cost, revenue, etc. It also discusses the difference between average and instantaneous rates of change. Several practice problems are provided relating to topics like rectilinear motion, fluid flow, population growth, and more.
Chapter 5
This document summarizes key concepts about uniform circular motion including:
- Radians are the SI unit for measuring angles where 1 radian is the central angle that spans an arc equal to the circle's radius.
- Formulas relate angular quantities like speed (ω) and displacement (θ) to linear quantities like speed (v) and arc length (s) using the radius (R).
- Centripetal force (Fc) is required to cause circular motion and is given by Fc = Mv2/R, where M is the object's mass and v is its speed.
- Banked roads allow vehicles to safely take curved portions faster by providing tilt that replaces needed friction with
VECTOR CALCULUS
The document summarizes Green's theorem, Stokes' theorem, and Gauss' divergence theorem from vector calculus. Green's theorem relates a line integral around a closed curve to a double integral over the region bounded by the curve. Stokes' theorem relates a surface integral over a closed surface to a line integral around its boundary. Gauss' divergence theorem relates the flux of a vector field through a closed surface to the volume integral of the divergence over the enclosed region. An example application of Gauss' theorem to compute the flux of a vector field out of a unit sphere is also provided.
5.13.6 Spheres and Composite Figures
This document provides information about calculating the surface area and volume of spheres, hemispheres, and composite figures. It defines spheres, hemispheres, and great circles. Formulas are provided for calculating the surface area and volume of spheres and hemispheres. Examples show how to use the formulas to find the surface area and volume of spheres and hemispheres given specific measurements. The document notes that composite figures can be broken down into simpler shapes to calculate their total surface area and volume by combining the surface areas and volumes of each individual shape. An example demonstrates this process for a composite prism and cylinder figure.
Answers assignment 4 real fluids-fluid mechanics
1) The document provides calculations to determine dynamic similitude between a model submarine and full-scale prototype based on Reynolds number. It is found that the prototype would need to operate at an unrealistically low speed of 0.044 m/s.
2) Additional calculations determine the corresponding prototype force of 7.4 N would be required for kinematic similarity.
3) Further calculations determine the prototype velocity and propulsion force required for surface propulsion based on Froude number, finding a velocity of 48 m/s but propulsion force of 6.2 MN, which is also unrealistic.
Presentation 3 ce801 by Rabindra Ranjan Saha, PEng, Assoc. Prof. WUB
The document discusses flow properties in open channels including:
- The Reynolds number and Froude number, which characterize flow regimes as turbulent or laminar and subcritical/supercritical.
- Hydraulic properties such as depth, area, wetted perimeter, hydraulic radius, and section factor which describe channel geometry.
- Critical flow occurs when the Froude number equals 1. Subcritical flow has a Froude number less than 1 while supercritical flow has a Froude number greater than 1.
- Examples are provided to demonstrate calculating hydraulic properties for given channel cross sections.
Ai4 heuristic2
The document discusses AND/OR graphs and the AO* algorithm for searching AND/OR trees. Some problems can be represented as having subgoals that can be achieved simultaneously or independently (AND) or as OR options. The AO* algorithm extends A* search to AND/OR trees. It examines multiple nodes simultaneously, selecting the most promising path and expanding nodes to generate successors. It computes heuristic values (h) for nodes and propagates new information up the graph as the search progresses until a solution is found or all paths are determined to be unsolvable. An example demonstrates how AO* searches an AND/OR graph and labels nodes as it proceeds.
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