Consider an arbitrary curve of fixed length C that connects the origin and a point a...
Consider an arbitrary curve of fixed length C that connects the origin and a point a fixed distance L < C away on the x-axis in the Cartesian x − y plane. Use the method of Lagrange multipliers to show that the area between the curve and the x-axis will be maximized if the curve forms the arc of a circle. (This shows, by the way, that a circle is the shape that encloses the largest area in a plane for a given fixed perimeter.)
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Consider an arbitrary curve of fixed length C that connects the
origin and a point a...
Assignment 4: (Arc Length and Surface Area - 7.3) 1. Consider the plane curve C defined...
Assignment 4: (Arc Length and Surface Area - 7.3) 1. Consider the plane curve C defined by y=e" between y-1 and y-e. (a.) Set up, but do NOT evaluate, an integral with respect to y for the arc length of C. (b.) Set up, but do NOT evaluate, an integral with respect to x for the arc length of C. Set up, but do NOT evaluate, an integral for the area of the surface obtained by rotating C about the...
An asteroid is particular mathematical curve: a hypocycloid with four cusps. It is formed by rolling a circle (show...
An asteroid is particular mathematical curve: a hypocycloid with four cusps. It is formed by rolling a circle (shown in black) inside a fixed circle (shown in blue) with four time the radius. The trajectory of an asteroid is shown in Figure1. Figure 1: Trajectory of an asteroid If the radius of the fixed circle is a, then the equation of an asteroid is given by Equation of an asteroid that needs to considered for this project is a. Use...
please solve it using "Cartesian method" An asteroid is particular mathematical curve: a hypocycloid with four...
please solve it using "Cartesian method"
An asteroid is particular mathematical curve: a hypocycloid with four cusps. It is formed by rolling a circle (shown in black) inside a fixed circle (shown in blue) with four time the radius. The trajectory of an asteroid is shown in Figure Figure 1: Trajectory of an asteroid If the radius of the fixed circle is a, then the equation of an asteroid is given by 2/3 2/3 2/3 Equation of an asteroid that...
An asteroid is particular mathematical curve: a hypocycloid with four cusps. It is formed by rolling a circle (sho...
An asteroid is particular mathematical curve: a hypocycloid with four cusps. It is formed by rolling a circle (shown in black) inside a fixed circle (shown in blue) with four time the radi trajectory of an asteroid is shown in Figure 1 us. The Figure 1: Trajectory of an asteroid Figure 1: Trajectory of an asteroid If the radius of the fixed circle is a, then the equation of an asteroid is given by 2/3 2/32/3 Equation of an asteroid...
a) Consider a straight conductor (infinite length) which carries a total curremt I in the direc-...
a) Consider a straight conductor (infinite length) which carries a total curremt I in the direc- tion of the z axis. Calculate the magnetic induction B(7) using the Biot-Savart law. Hint The current density is given by j = I8(x)6(y)ë^ b) Explicitly check the result from the previous task by calculating the magnetic induction B(F) using the symmetry of the problem and applying Stoke's theorem onto Maxwell's equation of magnetostatics V x B = HoJ. c) Now, consider a second...
3. Infinite conductor a) Consider a straight conductor (infinite length) which carries a total current I...
3. Infinite conductor a) Consider a straight conductor (infinite length) which carries a total current I in the direc- tion of the z axis. Calculate the magnetic induction Ē(7) using the Biot-Savart law. Hint The current density is given by j = 1S(x)6(y)ë^ b) Explicitly check the result from the previous task by calculating the magnetic induction B(F) using the symmetry of the problem and applying Stoke's theorem onto Maxwell's equation of magnetostatics V x B = 407 c) Now,...
question 4-7 4. Travelling Waves and Their Characteristics A rope wave travels in the positive x -direction. You are also told that the speed of the wave is 1000 cm/s, its frequency is 200 H...
question 4-7
4. Travelling Waves and Their Characteristics A rope wave travels in the positive x -direction. You are also told that the speed of the wave is 1000 cm/s, its frequency is 200 Hz, and that the wave is subject to the following initial conditions: at x 0 and t = 0: y =-1 cm, and, at x = 0 and t : ar = +20 cm/s (this is the velocity of the point on the rope at horizontal...
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3)....
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
Consider 1-2 Vr? + y + 3 LLL da dydar. V1-38-98 V +y + y2 +22...
Consider 1-2 Vr? + y + 3 LLL da dydar. V1-38-98 V +y + y2 +22 +y +22-2 the origin to the point (2, y, ) makes with the z-axis is a new angle which we will label o, and we label the length of the line segment p. We can now determine the remaining side-lengths of our new triangle. Let us try to label our point (2, y, z) in only p and 6. Our labeled triangle gives us...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
Assignment 4: (Arc Length and Surface Area - 7.3) 1. Consider the plane curve C defined by y=e" between y-1 and y-e. (a.) Set up, but do NOT evaluate, an integral with respect to y for the arc length of C. (b.) Set up, but do NOT evaluate, an integral with respect to x for the arc length of C. Set up, but do NOT evaluate, an integral for the area of the surface obtained by rotating C about the...
An asteroid is particular mathematical curve: a hypocycloid with four cusps. It is formed by rolling a circle (shown in black) inside a fixed circle (shown in blue) with four time the radius. The trajectory of an asteroid is shown in Figure1. Figure 1: Trajectory of an asteroid If the radius of the fixed circle is a, then the equation of an asteroid is given by Equation of an asteroid that needs to considered for this project is a. Use...
please solve it using "Cartesian method"
An asteroid is particular mathematical curve: a hypocycloid with four cusps. It is formed by rolling a circle (shown in black) inside a fixed circle (shown in blue) with four time the radius. The trajectory of an asteroid is shown in Figure Figure 1: Trajectory of an asteroid If the radius of the fixed circle is a, then the equation of an asteroid is given by 2/3 2/3 2/3 Equation of an asteroid that...
An asteroid is particular mathematical curve: a hypocycloid with four cusps. It is formed by rolling a circle (shown in black) inside a fixed circle (shown in blue) with four time the radi trajectory of an asteroid is shown in Figure 1 us. The Figure 1: Trajectory of an asteroid Figure 1: Trajectory of an asteroid If the radius of the fixed circle is a, then the equation of an asteroid is given by 2/3 2/32/3 Equation of an asteroid...
a) Consider a straight conductor (infinite length) which carries a total curremt I in the direc- tion of the z axis. Calculate the magnetic induction B(7) using the Biot-Savart law. Hint The current density is given by j = I8(x)6(y)ë^ b) Explicitly check the result from the previous task by calculating the magnetic induction B(F) using the symmetry of the problem and applying Stoke's theorem onto Maxwell's equation of magnetostatics V x B = HoJ. c) Now, consider a second...
3. Infinite conductor a) Consider a straight conductor (infinite length) which carries a total current I in the direc- tion of the z axis. Calculate the magnetic induction Ē(7) using the Biot-Savart law. Hint The current density is given by j = 1S(x)6(y)ë^ b) Explicitly check the result from the previous task by calculating the magnetic induction B(F) using the symmetry of the problem and applying Stoke's theorem onto Maxwell's equation of magnetostatics V x B = 407 c) Now,...
question 4-7
4. Travelling Waves and Their Characteristics A rope wave travels in the positive x -direction. You are also told that the speed of the wave is 1000 cm/s, its frequency is 200 Hz, and that the wave is subject to the following initial conditions: at x 0 and t = 0: y =-1 cm, and, at x = 0 and t : ar = +20 cm/s (this is the velocity of the point on the rope at horizontal...
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
Consider 1-2 Vr? + y + 3 LLL da dydar. V1-38-98 V +y + y2 +22 +y +22-2 the origin to the point (2, y, ) makes with the z-axis is a new angle which we will label o, and we label the length of the line segment p. We can now determine the remaining side-lengths of our new triangle. Let us try to label our point (2, y, z) in only p and 6. Our labeled triangle gives us...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
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