ESO 209: Assignment 3

August 18, 2015

PROBLEM 1

An electric motor must be arranged to turn a �ood control valve through an angle θ < π in a

right hand sense about a �xed directed line through the origin and the point P =(1/3, 2/3, 0

)m in

frame {E , ei}. An engineer has shown that any design is acceptable for which R11 = 3/5, R22 = 9/10and R33 = 1/2 are satis�ed in E . Determine the angle and the axis of rotation, and compute the

remaining six components of R in E .

PROBLEM 2

A cardboard box is initially positioned with its centroid C at the place(4/3, 2, 7/3

)m in frame

{E , ei} �xed in a packaging machine. The empty carton must be transported by a rotation about

a line through the point P =(2/3, 0,−1/3

)m in E to a new con�guration where the box is �lled

and sealed. The rotation required for the operation is speci�ed by the rotation matrix

[R]E =

1

9

4√3 + 1 −(1 +

√3) 5−

√3

5−√3 (1/2)(5

√3 + 8) (1/2)(5− 4

√3)

−(1 +√3) (1/2)(11− 4

√3) (1/2)(5

√3 + 8)

.Determine the rotation angle, and �nd the equation in E of the line of rotation. What is the dis-

placement of C? Find the location from the old location of C of the point T which initially was at

the top of the carton at the place(4/3, 2, 3

)m in E .

PROBLEM 3

A rigid body is rotated about a line through the point C at(1/3, 1, 5/3

)m in {E , ei}. The rotation

tensor in E is [R]E =

√2/2

√2/4

√6/4

0√3/2 −1/2

−√2/2

√2/4

√6/4

.Determine the �nal position in E of the particle P whose place initially was x = (

√2/3)e1 +

(2√3/9)e2 + (2/3)e3 m. What is the displacement of P? Find the axis and the angle of rotation of

the body in E .

1