Maple AD2016 4 answers rev1
Advanced Dynamics (WB2630 T1 S)
Technische Universiteit Delft
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Advanced Dynamics MapleTA test, week 4, printed version
TU Delft
October 16, 2016
1 Important information
In this test, you can fill in your answer as an algebraic calculation of numbers, for example 10
√
2
can be filled in as: 25(2)∧(1/2) or 25sqrt(0) or 20*sin(π/4) or 14.
Be sure to use a decimal point not a comma! So fill in 0 instead of 0,333 if you want to fill in decimal number version of 1/3. Your numerical answer is allowed to have a small rounding off error of±2% compared to the true answer.
The MapleTA test of this week consists of 17 questions. 12 of those are about Angular Velocity as discussed in the lectures on September 19thand September 22th. 5 questions are about Work and Energy as discussed in the lectures on September 26thand September 29th. They are mostly ordered, such that questions that require calculations aretowards the end of the test, and thus, the subjects of bot weeks are mixed. Nevertheless, use your time wisely, and don’t get stuck on questions you don’t know how to answer.
In this test, Matlab may be used. The questions marked as BONUS are more difficult and are the final questions. These questions are optional: If you do not make them, you can still score a grade 10.
For the printed version, three additional comments apply:
You have to mark your answers on the separate answer sheet youare provided with.
Fill out your name and studentID on the corresponding placeson the answer sheet. Check whether you filled in your studentID correctly.
Keep the answer sheet as clean as possible. A computer scans these sheets for automatic grading. Make sure you mark the answers clearly, and do not write on the sheet otherwise.
Check whether you filled in your studentID correctly again. The scanning algorithm relies on you having done this correctly!
1
2 Questions
- (1 point) Consider a homogeneous rigid body with one of itsbody-fixed axes coinciding with the body’s symmetry axis. The body-fixed axes are associated with frameB. What statement(s) is/are true about the inertia tensor expressed in frameB? Multiple answers can be selected. A. The tensor always is diagonal B. The tensor always is invertible C. The tensor always is symmetric D. The tensor always contains only the principal moments of inertia
- (1 point) A football with a perfect spherical shape is placed in the middle of the field prior to the second half, with its centroid at the exact location asit was placed prior to the first half. What is the least amount of points on the surface of the ball to be possibly in the same position as prior to the first half?Hint: Consider Euler’s axis of rotation theorem. A. 0 B. 1 C. 2 D. infinitely many
- (1 point) Four students are calculating the accelerationvectoraof a particle which rotates about an axis in a circular fashion with varying speed. The students are asked to give their answer expressed in the natural coordinate system. The parametera 0 is the magnitude of the acceleration vectoraatt = 0 s. Without further knowledge of the problem, which answer is definitelyincorrect?Multiple answers can be selected
NOTE:Dat antwoord ”D” niet kan was misschien niet helemaal duidelijk voor iedereen, gezien er wel een eenheid ontbrak, maar de numerieke waarde ook 0 was. Dit heeft misschien sommige mensen verward. Daarom hebben wij besloten om hier ook antwoorden goed te keuren waar optie ”D” ook aangevinkt was. A= 5eˆtm/s 2 B= 13 a 0 ˆen C= 5a 0 eˆn+ 3a 0 eˆb D=
[
0 0 0
]T
- (1 point) Consider a rotating coordinate systemxyz, in which a particle moves relative to its axes. The rotating coordinate system is associated withframeBand rotates relative to coordinate systemXYZ, which is associated with frameNwith vectorNω=
[
ω 1 ω 2 ω 3
]T
.
The origins of both coordinate systems coincide at all times. Expressed in the inertial frame N, the particle’s position with respect to the origin is denoted with vectorNr=aˆex+bˆey, whereaandbare time-dependent scalar distances. How can the particle’s velocity correctly be expressed in the inertial frame?Multiple answers can be selected. A= ̇aeˆx+b ̇ˆey B= ̇aeˆX+b ̇eˆY C= ( ̇a−bω 3 )eˆx+ (b ̇+aω 3 )ˆey+ (ω 1 b−ω 2 a)eˆz D= ̇aeˆx+aˆe ̇x+b ̇eˆy+beˆ ̇y
- (1 point) Consider a thin rolling wheel with a body-fixed coordinate systemxyz, associated with frameB. The originO’of the coordinate system lies in the centroid of the wheel. The z-axis is aligned with the axle of the wheel and thex- andy-axes are fixed to the spokes (Dutch: ”spaken”). Let the wheel roll over the ground without slipping. Movements are considered relative to the inertial coordinate systemXYZ, associated with frameN, where theXZ-plane is the ground. The wheel is constrained, such that it can rotate about theY- axis and itsz-axis. To which axis/axes is the instantaneous axis of rotation perpendicular? Multiple answers can be selected
NOTE: Omdat het niet duidelijk vermeld is dat het ook mogelijk is dat er 0 antwoorden juist kunnen zijn, heeft iedereen een punt voor deze vraag gekregen. A. always to thex-axis B. always to they-axis C. always to thez-axis D. always to theZ-axis
- (1 point) Consider a person in a rotating ride in an amusement park with body-fixed coor- dinate systemxyz, associated with frameBand originO’, rotating around theZ-axis of an inertial coordinate systemXYZ, which is associated with frameNand originO. The person rotates with a constant angular velocity of magnitudeω 1. A bee flies relative to the person and the bee’s position is defined by the position vectorBρ, with respect toO′. Assume that att = 0 s, the directional unit vectors of bothBandN frames are aligned and that the origins of both coordinate systemsOandO’coincide at all times. Att=0 s, determine theY-componentvY of the velocityBvof the bee, relative to the inertial frameN, expressed in the body-fixed frameB. Use:
- Position vectorBρ= 3 sin (ω 2 t)ˆexm+4 cos (ω 2 t)eˆym, whereω 2 is a constant angular velocity.
- Angular velocities: ω 1 = 2πrad/s andω 2 =πrad/s.
Solution: 2 πm/s
- (1 point) Consider again a person in a rotating ride in an amusement park with body-fixed coordinate systemxyz, associated with frameBand originO′, rotating around theZ-axis of an inertial coordinate systemXY Z, which is associated with frameNand originO. The person rotates with a constant angular velocity of magnitudeω 1. A bee flies relative to the person and the bee’s position is defined by the position vectorBρ, with respect toO′. Assume that att= 0 s, the directional unit vectors of bothBandNframes are aligned and that the origins of both coordinate systemsOandO′coincide at all times. Att=0 s, determine theY-componentaY of the accelerationBaof the bee, relative to the inertial frameN, expressed in the body-fixed frameB. Use:
- Position vectorBρ= 3 sin (ω 2 t)ˆexm +4 cos (ω 2 t)eˆym, whereω 2 is a constant angular velocity.
- Angular velocities: ω 1 = 2πrad/s andω 2 =πrad/s.
Solution:0 m/s 2
- (1 point) A particle rotates in a circular fashion aroundtheZ-axis of a coordinate system, which is associated with inertial frameNwith angular velocity of magnitudeω, and simul- taneously moves in the positive direction of that axis with speedZ ̇. Give theZ-component of the time derivative of the tangential unit vectorˆetif:
- ω= 4 rad/s,
- Z ̇= 0 m/s and
- the circle has a radius ofr= 3 m.
Solution:0 s− 1
- (1 point) BONUS: In an inertial ENU-coordinate system, atrain accelerates in the East- direction with magnitudea. A passenger travels with his back towards the driving direction and holds a spinning wheel sideways using a rod going throughthe axle, such that the axis about which the wheel rotates points in the Up-direction. Around one of the spokes (Dutch: ”spaken”) of the wheel, a coloured marble is attached and starts moving at timet=0s. If:
- The marble is moving outward from the center towards the rim (Dutch: ”buitenrand”) of the wheel with a speed defined byv=bt,
- parameterbis a constant acceleration ofb= 0 m/s 2 relative to the wheel’s center,
- the train accelerates witha= 2 m/s 2
- the wheel spins with a constant angular velocity of magnitudeω=πrad/s,
determine the magnitude of the acceleration of the marble, relative to the inertial coordinate system att= 0 s. At this time, the marble is located 10 cm from the center of the wheel, pointing to the inertial North direction.
Solution:2 m/s 2
- (1 point) BONUS: Consider a rigid body with a body-fixed coordinate systemxyzwith the origin in its center of mass, associated with frameB. The body is first rotated about its y-axis, then about itsxaxis and finally about itsy-axis again (which now has a different orientation than before!). Construct a symbolic rotation matrix which describes the total
rotation and compare this with the given rotation matrixBCN=
− 120 −
√ 3 2 −
√ 3 4
√ 3 2
1 4 3 4
1 2 −
√ 3 4
.
What angle was the rotation about thex-axis? Give your answer within 0≥θ≥πrad or 0 ≥θ≥180 deg.
Solution:π/6 rad
- (1 point) BONUS: Consider again a thin rolling wheel witha body-fixed coordinate system xyz, associated with frameBand originO’. Thez-axis passes through the centred axle and thex- andy-axes are fixed to the spokes. Let the wheel roll over the ground without slipping. Movements are considered relative to the inertial coordinate systemXYZ, associated with frame N and origin O, where theXZ-plane is the ground. The wheel now can rotate around all axes wheel is rolling in a perfect circular motion about the positiveY axis, completing one cycle with radiusR, fromO to the contact point of the wheel with the ground. Consider the wheel in the following orientation:
- Itsx-axis is parallel to the ground, pointing into the directionof where the axle moves to.
- The wheel is tilted about itsx-axis, such that itsz-axis points towardsO.
- The unit vector in thez-direction has noX-component.
Give the componentωZof angular velocity vectorNωof the wheel, as expressed in theN frame if:
- the wheel’s radius isr= 1 m,
- the circle around theY-axis has radiusR =5 m,
- one revolution about theY-axis is done in 20 seconds.
Solution:± 1 .5391 rad/s
Maple AD2016 4 answers rev1
Vak: Advanced Dynamics (WB2630 T1 S)
Universiteit: Technische Universiteit Delft
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