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5. (3 pts) Any operator that transfors the same way as the position operator r under rotation is ...
5. (3 pts) Any operator that transfors the same way as the position operator r under rotation is called a vector operator. By "transforming the same way" we mean that V DV where D is the same matrix as appears in Dr. In particular for a rotation about the z axis we should have cos p-sinp0 sincos 0 0 where φ is the angle of rotation. This transformation rule follows frorn the generator of rotations where n is the unit...
correct answers 135. Computer Graphics. One of the most important applications of linear transformations is...
correct answers
135. Computer Graphics. One of the most important applications of linear transformations is computer graphics where we wish to view 3-dimensional objects (for example a crystal) on a 2-dimensional screen. The screen is the ry-plane. The aim is to rotate the crystal and orthogonally project it onto the ry-plane to obtain different views of it. We consider 3 possible rotations: 0 . A rotation of θ round the x-axis using the matrix R2-10 cos θ -sin θ cos...
3 Angular Momentum and Spherical Harmonics For a quantum mechanical system that is able to rotate...
3 Angular Momentum and Spherical Harmonics For a quantum mechanical system that is able to rotate in 3D, one can always define a set of angular momentum operators J. Jy, J., often collectively written as a vector J. They must satisfy the commutation relations (, ] = ihſ, , Îu] = ihſ, J., ſu] = ihỈy. (1) In a more condensed notation, we may write [1,1]] = Žiheikh, i, j= 1,2,3 k=1 Here we've used the Levi-Civita symbol, defined as...
6Show that the Eulerian strain rate is given by 147 and [see Eq. (3.5.10) for the definition of a...
6Show that the Eulerian strain rate is given by 147 and [see Eq. (3.5.10) for the definition of al e D. 3.5.3 Infinitesimal Rotation Tensor The displacement gradient tensor can be expressed as the sum of tensor and a skew symmetric tensor. We have of a symmetric where the symmetric part is similar to the infinitesimal strain tensor (and when l▽u| ~ |▽oul << 1), and the skew symmetric part is known as the infinitesimal rotation tensor lelilt lh the...
2. For the following problems, indicate if the statement is TRUE or FALSE. If the statement...
2. For the following problems, indicate if the statement is TRUE or FALSE. If the statement is FALSE, state why and make necessary corrections. [13 points total] 1. If two wave functions are orthogonal, then they are also both normalized. b. If a system is in a state described by a wavefunction Y, then the average value of the observable corresponding to the operator A is given by the equation YAY dt (a) [ 'ዋ dr c. The momentum operator,...
FBD Use Sum M = r x mv - r x mv (final momentum - initial...
FBD
Use Sum M = r x mv - r x mv (final momentum - initial momentum) and
polar coordinates: e sub theta, e sub r, and k directions
2) On top of a baby's crib there is a sun and moon mobile. The motor provides a 0.03 N-cm torque about the j-axis and has a terminal angular velocity of 35 RPM (rotations per minute). The moon and the sun both have a mass of 25 grams. The position vector...
In the 3D Cartesian system the rotation matrix is around the z-axis is (a 2D rotation):...
In the 3D Cartesian system the rotation matrix is around the
z-axis is (a 2D rotation):
Where
is the angle to rotate. Then rotation from A to A' is can be
represented via matrix multiplications: [A'] = [R][A]
Such a rotation is useful to return a system solved in
simplified co-ordinates to it's original co-ordinate system,
returning to original meaning to the answer. A full 3D rotation is
simply a series of 2D rotations (with the appropriate matrices)
Question: If...
Heres example 10.2 (3) (30 points) In Example 10.2, the moment of inertia tensor for a...
Heres example 10.2
(3) (30 points) In Example 10.2, the moment of inertia tensor for a uniform solid cube of mass Mand side a is calculated for rotation about a corner of the cube. It also worked out the angular momentum of the cube when rotated about the x-axis - see Equation 10.51. (a) Find the total kinetic energy of the cube when rotated about the x-axis. (b) Example 10.4 finds the principal axes of this cube. It shows that...
Problem: Given a rotation R of R3 about an arbitrary axis through a given angle find the matrix w...
Problem: Given a rotation R of R3 about an arbitrary axis through a given angle find the matrix which represents R with respect to standard coordinates. Here are the details: The axis of rotation is the line L, spanned and oriented by the vector v (1,一1,-1) . Now rotate R3 about L through the angle t = 4 π according to the Right 3 Hand Rule Solution strategy: If we choose a right handed ordered ONB B- (a, b,r) for...
PLEASE SHOW WORK AND ANSWER ALL PARTS. WHEN I ASKED THE SECTIONS SEPARATELY THE ANSWERS DID NOT MATCH, THANK YOU!!! All of the questions on this concern a space station, consisting of a long thin unif...
PLEASE SHOW WORK AND ANSWER ALL PARTS. WHEN I ASKED THE
SECTIONS SEPARATELY THE ANSWERS DID NOT MATCH, THANK
YOU!!!
All of the questions on this concern a space station, consisting
of a long thin uniform rod of mass 4.4 x
106kg and length 171 meters,
with two identical uniform hollow spheres, each of mass 1.3
x 106 kg and radius 57 meters,
attached at the ends of the rod, as shown below. Note that none
of the diagrams shown...
5. (3 pts) Any operator that transfors the same way as the position operator r under rotation is called a vector operator. By "transforming the same way" we mean that V DV where D is the same matrix as appears in Dr. In particular for a rotation about the z axis we should have cos p-sinp0 sincos 0 0 where φ is the angle of rotation. This transformation rule follows frorn the generator of rotations where n is the unit...
correct answers
135. Computer Graphics. One of the most important applications of linear transformations is computer graphics where we wish to view 3-dimensional objects (for example a crystal) on a 2-dimensional screen. The screen is the ry-plane. The aim is to rotate the crystal and orthogonally project it onto the ry-plane to obtain different views of it. We consider 3 possible rotations: 0 . A rotation of θ round the x-axis using the matrix R2-10 cos θ -sin θ cos...
3 Angular Momentum and Spherical Harmonics For a quantum mechanical system that is able to rotate in 3D, one can always define a set of angular momentum operators J. Jy, J., often collectively written as a vector J. They must satisfy the commutation relations (, ] = ihſ, , Îu] = ihſ, J., ſu] = ihỈy. (1) In a more condensed notation, we may write [1,1]] = Žiheikh, i, j= 1,2,3 k=1 Here we've used the Levi-Civita symbol, defined as...
6Show that the Eulerian strain rate is given by 147 and [see Eq. (3.5.10) for the definition of al e D. 3.5.3 Infinitesimal Rotation Tensor The displacement gradient tensor can be expressed as the sum of tensor and a skew symmetric tensor. We have of a symmetric where the symmetric part is similar to the infinitesimal strain tensor (and when l▽u| ~ |▽oul << 1), and the skew symmetric part is known as the infinitesimal rotation tensor lelilt lh the...
2. For the following problems, indicate if the statement is TRUE or FALSE. If the statement is FALSE, state why and make necessary corrections. [13 points total] 1. If two wave functions are orthogonal, then they are also both normalized. b. If a system is in a state described by a wavefunction Y, then the average value of the observable corresponding to the operator A is given by the equation YAY dt (a) [ 'ዋ dr c. The momentum operator,...
FBD
Use Sum M = r x mv - r x mv (final momentum - initial momentum) and
polar coordinates: e sub theta, e sub r, and k directions
2) On top of a baby's crib there is a sun and moon mobile. The motor provides a 0.03 N-cm torque about the j-axis and has a terminal angular velocity of 35 RPM (rotations per minute). The moon and the sun both have a mass of 25 grams. The position vector...
In the 3D Cartesian system the rotation matrix is around the
z-axis is (a 2D rotation):
Where
is the angle to rotate. Then rotation from A to A' is can be
represented via matrix multiplications: [A'] = [R][A]
Such a rotation is useful to return a system solved in
simplified co-ordinates to it's original co-ordinate system,
returning to original meaning to the answer. A full 3D rotation is
simply a series of 2D rotations (with the appropriate matrices)
Question: If...
Heres example 10.2
(3) (30 points) In Example 10.2, the moment of inertia tensor for a uniform solid cube of mass Mand side a is calculated for rotation about a corner of the cube. It also worked out the angular momentum of the cube when rotated about the x-axis - see Equation 10.51. (a) Find the total kinetic energy of the cube when rotated about the x-axis. (b) Example 10.4 finds the principal axes of this cube. It shows that...
Problem: Given a rotation R of R3 about an arbitrary axis through a given angle find the matrix which represents R with respect to standard coordinates. Here are the details: The axis of rotation is the line L, spanned and oriented by the vector v (1,一1,-1) . Now rotate R3 about L through the angle t = 4 π according to the Right 3 Hand Rule Solution strategy: If we choose a right handed ordered ONB B- (a, b,r) for...
PLEASE SHOW WORK AND ANSWER ALL PARTS. WHEN I ASKED THE
SECTIONS SEPARATELY THE ANSWERS DID NOT MATCH, THANK
YOU!!!
All of the questions on this concern a space station, consisting
of a long thin uniform rod of mass 4.4 x
106kg and length 171 meters,
with two identical uniform hollow spheres, each of mass 1.3
x 106 kg and radius 57 meters,
attached at the ends of the rod, as shown below. Note that none
of the diagrams shown...
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