Physics 3210If v is a vector in one frame and v is the same vector in a rotated frame, we showed that v=Av, where A is a rotation matrix (the matrix of direction cosines). This determines v given A and v. If instead we know v and A, how can we find v?For a rotation matrix, ATA=I. What does this imply about the determinant of A?Which of the following is the correct matrix for a rotation about the x axis?Suppose a book is rotated in two different ways: from the same starting orientation, the book is rotated by 90o clockwise (1) about the x axis, then about the y axis, then about the z axis, or (2) about the y axis, then about the x axis, then about the z axis. Note: the axes are in the space frame. How do the final orientations of the book compare?Using Euler angles, we wish to construct a rotation matrix that rotates first by angle φ about the z axis (matrix Aφ), then by angle θ about the x axis (matrix Aθ), then by angle ψ about the z axis (matrix Aψ). How should we multiply the three rotation matrices to get the final rotation matrix, A(φ,θ,ψ)?Using Euler angles, we constructed a rotation matrix A(φ,θ,ψ) that rotates first by angle φ about the z axis (matrix Aφ), then by angle θ about the x axis (matrix Aθ), then by angle ψ about the z axis (matrix Aψ). What is the matrix A-1 that reverses (or undoes) this series of rotations?Slide 8What is the relationship between the principle moments of inertia I1, I2, and I3 for a symmetric top?For a symmetric top with I2=I3, we defined ε=(I1-I2)/I2.What is the sign of ε for an oblate object such as a coin?We showed that for force-free motion of a symmetric top, the angular momentum vector is given by What does this imply about the geometry of the three vectors?For force-free motion of a symmetric top in the body frame, ω precesses about the e1 axis with angular velocity Ω, and the angular momentum L lies in the same plane as ω and e1. What does this imply about the motion of the angular momentum vector L in the body frame?For force-free motion of a symmetric top in the space frame, we showed that . What does this imply about the motion?Slide 14Physics 3210Week 12 clicker questionsIf v is a vector in one frame and v is the same vector in a rotated frame, we showed that v=Av, where A is a rotation matrix (the matrix of direction cosines). This determines v given A and v.If instead we know v and A, how can we find v?A. v=AvB. v=ATvC. v=A2vD. We can’t determine v without first determining a different rotation matrix.For a rotation matrix, ATA=I. What does this imply about the determinant of A?A. det A = 1B. det A = -1C. det A = 1 or -1D. det A = 0E. None of the above.Which of the following is the correct matrix for a rotation about the x axis?xcos sin 0A sin cos 00 0 1q q� �� �= - q q� �� �� �A.B. x1 0 0A 0 sin cos0 cos sin� �� �= q q� �� �- q q� �C.D. x1 0 0A 0 cos sin0 sin cos� �� �= q q� �� �- q q� �xsin cos 0A cos sin 00 0 1q q� �� �= - q q� �� �� �Suppose a book is rotated in two different ways: from the same starting orientation, the book is rotated by 90o clockwise(1) about the x axis, then about the y axis, then about the z axis,or(2) about the y axis, then about the x axis, then about the z axis.Note: the axes are in the space frame.How do the final orientations of the book compare?A. The orientations are the same.B. The orientations are different.Using Euler angles, we wish to construct a rotation matrix that rotates first by angle φ about the z axis (matrix Aφ), then by angle θ about the x axis (matrix Aθ), then by angle ψ about the z axis (matrix Aψ). How should we multiply the three rotation matrices to get the final rotation matrix, A(φ,θ,ψ)?A. A(φ,θ,ψ)=AθAψAφ B. A(φ,θ,ψ)=AφAθAψ C. A(φ,θ,ψ)=AφAψAθ D. A(φ,θ,ψ)=AψAφAθ E. A(φ,θ,ψ)=AψAθAφUsing Euler angles, we constructed a rotation matrix A(φ,θ,ψ) that rotates first by angle φ about the z axis (matrix Aφ), then by angle θ about the x axis (matrix Aθ), then by angle ψ about the z axis (matrix Aψ). What is the matrix A-1 that reverses (or undoes) this series of rotations?A. A-1=A(φ,θ,ψ) B. A-1=AT(φ,θ,ψ) C. A-1=A(θ,φ,ψ) D. A-1=AT(θ,φ,ψ)E. A-1=A(ψ,φ,θ)Physics 3210Wednesday clicker questionsWhat is the relationship between the principle moments of inertia I1, I2, and I3 for a symmetric top?A. I1 = I2 = I3B. I1 ≠ I2 = I3C. I1 ≠ I2 ≠ I3D. The answer depends on the choice of axes.For a symmetric top with I2=I3, we defined ε=(I1-I2)/I2.What is the sign of ε for an oblate object such as a coin?A. ε>0B. ε=0C. ε<0D. The answer depends on the choice of axes.We showed that for force-free motion of a symmetric top, the angular momentum vector is given by What does this imply about the geometry of the three vectors?A. B. C. D. E. ( )1 2 1 1 2I I I= - w +L eω1, , and lie in the same plane.L eω1, , and are orthogonal to each other.L eω1 and are orthogonal to each other.L e and are orthogonal to each other.Lω1, , and are parallel.L eωFor force-free motion of a symmetric top in the body frame, ω precesses about the e1 axis with angular velocity Ω, and the angular momentum L lies in the same plane as ω and e1. What does this imply about the motion of the angular momentum vector L in the body frame?A. L is constant.B. L precesses about the ω axis with angular velocity Ω. C. L precesses about the ω axis an angular velocity different from Ω. D. L precesses about the e1 axis with angular velocity Ω. E. L precesses about the e1 axis an angular velocity different from Ω.For force-free motion of a symmetric top in the space frame, we showed that.What does this imply about the motion?A. The vector ω precesses about the L axis with angular velocity ωp =L/(2I2)B. The vector ω precesses about the L axis with angular velocity ωp =L/I2C. The vector L precesses about the ω axis with angular velocity ωp =L/I2D. The vector L precesses about the ω axis with angular velocity ωp =L/(2I2) E. The vector ω precesses about the e1 axis with angular velocity ωp =L/I22I= �Lω ω&Physics 3210Friday clicker