Coordinate system in rotation. Consider the Minkowski space in spherical coordinates (t, r, θ, φ) and perform a coordina...
Coordinate system in rotation.
Consider the Minkowski space in spherical coordinates (t, r, θ, φ)
and perform a coordinate transformation to a rotational system
given by t '= t, r' = r, θ '= θ, φ' = φ + ωt.
(a) Find the metric in the new coordinates and all the Christoffel symbols.
(b) Take θ' = θ = π/2. Write the equations of the geodesic and compare with the equations d²xi'/ dt'² = f^i, find the value of the "inertial forces" f^i. Is it possible to have a particle at rest in these coordinates (U^r' = U^φ' = 0) without applying an external force? If required, what is the external force that must be added to get the particle to remain at rest?
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Coordinate system in rotation. Consider the Minkowski space in spherical coordinates (t, r, θ, φ) and perform a coordina...
Consider a two-dimensional space with coordinates x µ = (θ, φ), for which the only nonvanishing...
Consider a two-dimensional space with coordinates x µ = (θ, φ),
for which the only nonvanishing Christoffel symbols are
3. Consider a two-dimensional space with coordinates Ζμ-(9,0), for which the only non- vanishing Christoffel symbols are tan θ Write down the two components of the geodesic equation, and find a set a solutions.
(10 marks) In class we had a question regarding the spherical coordinate system: Given that rcos θ sin φ y-rsin0 sin o with 0 θ 2π and 0 φ π "Why don't we have 0 θ π and 0 φ 2π instead...
(10 marks) In class we had a question regarding the spherical coordinate system: Given that rcos θ sin φ y-rsin0 sin o with 0 θ 2π and 0 φ π "Why don't we have 0 θ π and 0 φ 2π instead" (a) (5 marks) Explain why this would not work b) (5 marks) If you really wanted the bounds suggested how could you make it work?
(10 marks) In class we had a question regarding the spherical coordinate system:...
1.18. Points P and P' have spherical coordinates (r,0,y) and (r,θ,φ), cylindrical coordinates (p, p, z)...
1.18. Points P and P' have spherical coordinates (r,0,y) and (r,θ,φ), cylindrical coordinates (p, p, z) and (p',p',z'), and Cartesian coordinates (x, y, z) and (x',y',z'), respectively. Write r - r in all three coordinate systems. Hint: Use Equation 1.2) with a r r and r and r' written in terms of appropriate unit vectors.
Use a polar coordinate system to plot the point with the given polar coordinates. Then find another representation (r,θ) of this point in which: Use a polar coordinate system to plot the point wi...
Use a polar coordinate system to plot the point with the given
polar coordinates. Then find another representation (r,θ) of this
point in which:
Use a polar coordinate system to plot the point with the given polar coordinates. Then find another representation (r,0) of this point in which: (b) r< 0,0 <θ<2x. Choose the correct graph below. O A O B O C. O D. ピ -5 (a) What are the coordinates of the point for which r > 0,...
question 12 , please sketch it by your hand , do not use computer graph θ varies from 0 to 2 π. φ varies from 0 to π/4 while 0 is constant. find 9-10 Write the equation in spherical coordinates. 9. (...
question 12 , please sketch it by your hand , do not use
computer graph
θ varies from 0 to 2 π. φ varies from 0 to π/4 while 0 is constant. find 9-10 Write the equation in spherical coordinates. 9. (a) :2-x2 + y2 10. (a) a-2r+y- (b) x2 +z2 = 9 (b) x + 2y+ 3:-1 11-14 Sketch the solid described by the given inequalities. 15. A solid lies above the cone:- + y and below the sphere...
(a) In Cartesian coordinates, the unit vectors r and φ are related to the unit vectors...
(a) In Cartesian coordinates, the unit vectors r and φ are related to the unit vectors x and y by Using these expressions, and r-rr, derive Eqs. (1.48) for Fr and Fo: F, m(ř_rơ), = 1.48: (Unlike r and ф. æ and y do not change with time.) (b) Consider a particle that feels an angular force only, of the form Fo mro. (There is nothing physical about this force; it simply makes the F-ma equations solv- able.) Show that...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r,...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
(2, Consider the Laplace equation for a ball of radius R described in spherical coordinates (T,0) 2 1 +00n3 2 0 wher...
(2, Consider the Laplace equation for a ball of radius R described in spherical coordinates (T,0) 2 1 +00n3 2 0 where 0 is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w =cos e given by 12 F" (T) +2r F (r) - n(n + 1 ) Fr (r) = 0, (1 w2)G (w) - 2wG (w) + n(n+ 1)G, (w)...
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...
For the system shown in Fig. 1, solve the following problems. (a) Find the transfer function, G(s...
For the system shown in Fig. 1, solve the following problems. (a) Find the transfer function, G(s)X2 (s)/F(s) (b) Does the system oscillate with a unit step input (f (t))? Explain the reason (c) Decide if the system(x2 (t)) is stable with a unit step input (f (t))? Explain the reason 1. 320) 8 kg 2 N/m 4N-s/m 2N-s/m Fig. 1 2. There are two suspensions for a car as shown in Fig. 2 (a) Find the equations of each...
Consider a two-dimensional space with coordinates x µ = (θ, φ),
for which the only nonvanishing Christoffel symbols are
3. Consider a two-dimensional space with coordinates Ζμ-(9,0), for which the only non- vanishing Christoffel symbols are tan θ Write down the two components of the geodesic equation, and find a set a solutions.
(10 marks) In class we had a question regarding the spherical coordinate system: Given that rcos θ sin φ y-rsin0 sin o with 0 θ 2π and 0 φ π "Why don't we have 0 θ π and 0 φ 2π instead" (a) (5 marks) Explain why this would not work b) (5 marks) If you really wanted the bounds suggested how could you make it work?
(10 marks) In class we had a question regarding the spherical coordinate system:...
1.18. Points P and P' have spherical coordinates (r,0,y) and (r,θ,φ), cylindrical coordinates (p, p, z) and (p',p',z'), and Cartesian coordinates (x, y, z) and (x',y',z'), respectively. Write r - r in all three coordinate systems. Hint: Use Equation 1.2) with a r r and r and r' written in terms of appropriate unit vectors.
Use a polar coordinate system to plot the point with the given
polar coordinates. Then find another representation (r,θ) of this
point in which:
Use a polar coordinate system to plot the point with the given polar coordinates. Then find another representation (r,0) of this point in which: (b) r< 0,0 <θ<2x. Choose the correct graph below. O A O B O C. O D. ピ -5 (a) What are the coordinates of the point for which r > 0,...
question 12 , please sketch it by your hand , do not use
computer graph
θ varies from 0 to 2 π. φ varies from 0 to π/4 while 0 is constant. find 9-10 Write the equation in spherical coordinates. 9. (a) :2-x2 + y2 10. (a) a-2r+y- (b) x2 +z2 = 9 (b) x + 2y+ 3:-1 11-14 Sketch the solid described by the given inequalities. 15. A solid lies above the cone:- + y and below the sphere...
(a) In Cartesian coordinates, the unit vectors r and φ are related to the unit vectors x and y by Using these expressions, and r-rr, derive Eqs. (1.48) for Fr and Fo: F, m(ř_rơ), = 1.48: (Unlike r and ф. æ and y do not change with time.) (b) Consider a particle that feels an angular force only, of the form Fo mro. (There is nothing physical about this force; it simply makes the F-ma equations solv- able.) Show that...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
(2, Consider the Laplace equation for a ball of radius R described in spherical coordinates (T,0) 2 1 +00n3 2 0 where 0 is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w =cos e given by 12 F" (T) +2r F (r) - n(n + 1 ) Fr (r) = 0, (1 w2)G (w) - 2wG (w) + n(n+ 1)G, (w)...
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...
For the system shown in Fig. 1, solve the following problems. (a) Find the transfer function, G(s)X2 (s)/F(s) (b) Does the system oscillate with a unit step input (f (t))? Explain the reason (c) Decide if the system(x2 (t)) is stable with a unit step input (f (t))? Explain the reason 1. 320) 8 kg 2 N/m 4N-s/m 2N-s/m Fig. 1 2. There are two suspensions for a car as shown in Fig. 2 (a) Find the equations of each...
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