dynamics - retake exam 27.08.2018- type 1 - n.ethz.chkiranku/dynamics2018/files/dynamics-final exam...

Dynamics - Retake Exam 27.08.2018- Type 1

1. Consider a particle subjected to a holonomic constraint. Which of the followingis true?

(a) A holonomic constraint depends on the current position of the particle anddepends on time-derivatives of the motion of the particle.

(b) A holonomic constraint is a constraint that only depends on time.

(c)I A holonomic constraint is a constraint that can depend on the current posi-tion of the particle and time.

(d) A holonomic constraint is a constraint that only depends on the currentposition of the particle.

(e) A holonomic constraint depends on time-derivatives of the motion of theparticle.

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2. Consider the one-degree-of-freedom forced-damped oscillator: x + 0.2x + x =sin(ωt) and its particular solution xp(t) = A sin(ωt − ϕ). What is the correctpicture representing the diagrams of the amplitude A and the phase shift ϕ?

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The correct answer is (d).

2

3. The planar multibody system shown in thefigure consists of 5 rigid bars L1, L2, L3,L4, L5 and a rigid disk B. The bodies L1,L2, L5 and B are hinged to the ground atthe points A, B, H and G, respectively.Furthermore, L1 and L2 are hinged to L3

at C and D, while L5 and B are hinged toL4 at E and F . Moreover, a spring of stiff-ness k connects points E and C, as shown,thus linking the bodies L1, L3, L4 and L5.How many degrees of freedom does the sys-tem have?

(a) 4

(b) 3

(c) 1

(d) 0

(e)I 2

3

4. The system shown in the figure lies in the hor-izontal plane containing a rigid guide R alongwhich a rigid body C can slide without friction.The guide is hinged to the ground at point H.A massless rigid rod with two hinges at theextremities connects the fixed point O to thecenter of mass of C in S. The angle enclosed byOS and the e1 axis of the fixed reference frame[e1, e2] is denoted by θ. At the time instant

depicted, θ = π/3 and the angle OHS = π/2.What is the velocity c, i.e. the time derivativeof the length HS?

(a) c = −cθ(√

3 +√

6)

(b) c = cθ√

2

(c) c = −2c√θ

(d)I c = −cθ√

3

(e) c = −3θ2√

3

4

mm

m5. Three small spheres of equal mass m are con-nected by a spring of elastic constant k andby a massless rigid link, as shown. The wholesystem moves on a horizontal, frictionless pla-nar surface. If a force F is a applied to onemass as shown, what is the acceleration of thecenter of mass of the spheres aCM at the in-stant depicted in the figure? Assume that thestretching of the spring is given by x(t).

(a) aCM = − 1

mF

(b)I aCM =1

3mF

(c) aCM =1

3mk(F + kx(t))

(d) aCM =k

2mx(t) +

1

mF

(e) aCM =1

3mF + x(t)

5

!

HA

A

B

ex

ey

ez

6. Consider a rigid body B and an arbitrarypoint A on B. The body has an angu-lar velocity ω and an angular momentumHA w.r.t. point A. The coordinate frame[ex ey ez] is centered in A and I

Ais the

moment of inertia tensor of B w.r.t. thisframe. If the angular momentum can bewritten as HA = I

Aω, which of the follow-

ing statements must be true?

(a) No forces are applied to the body.

(b) ω and HA are parallel.

(c)I Point A is either fixed or coincideswith the center of mass.

(d) IA

is diagonal.

(e) Point A is fixed and coincides with the center of mass.

7. The object depicted in the figure is obtained by removing a circle of radius a/2 froma rectangle of sides 2a and 2b and of unit planar density ρ ([ρ] = [kg/m2]). What isthe moment of inertia IO of the object w.r.t pointO? Hint: the centroidal momentof inertia ID of a rectangle of sides w and h and mass m is I = 1

12m(w2 + h2).

The centroidal moment of inertia of a disk of radius r and mass m is I = 12mr2.

aa

b

b

O

a/2

(a) IO =4

3ρa3b+

12

3ρab3 +

ρπa2

4

(b2 +

1

2a2)

(b) IO =2

3ρab3 +

12

3ρa3b− ρπa2

4

(b2 +

1

8a2)

(c) IO =1

3ρa3b+

19

3ρab3 − ρπa2

4

(b+

1

2a2)

(d) IO =2

3ρa2b2 +

16

3ρa2b2 − ρπa2

4

(b2 +

1

8a2)

(e)I IO =4

3ρa3b+

16

3ρab3 − ρπa2

4

(b2 +

1

8a2)

6

mA

mB

a

b

ex

ey

O

vA

vB

mC

vC

F

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8. The system shown in the figure consists of threeparticles of mass mA, mB and mC , respectively,connected by a massless, rigid frame. The dimen-sions of the frame are shown in the figure, denotedby the parameters a and b. At the instant de-picted, vA = vex + vey and vB = vex + 2vey. Aforce F = −Fex is applied on mC . What is thevelocity vC?

(a) vC = b2avex + 3

4vey

(b)I vC =(1− 2

3ba

)vex + 4

3vey

(c) vC = −√

FbmA+mB+mC

ex

(d) vC =√

Fa2

b(mA+mB+mC)ex

(e) vC =(1 + 2b

a

)vex + 1

2vey

7

mA

mB

a

b

O

MO

!0

9. The system shown in the figure consists of two par-ticles of mass mA and mB, respectively, connectedby a massless rigid bar. The system is hinged atpoint O and rotates about it with constant angu-lar velocity ω0 up until time t = 0. The distancesbetween O and the two particles are indicated bya and b, respectively. Starting from time t = 0,a torque of magnitude M(t) = M0t, where M0 isconstant, is applied at the joint O until the systemcomes to a halt at time t = T . What is the valueof T? Ignore gravity.

(a) T →∞

(b) T =√ω0

mBa2+mAb2

2M0

(c)I T =√

2ω0mBb2+mAa2

M0

(d) T = 0

(e) T = ω0mBb

2+mAa2

M0ab

8

g

m O1

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10. Two disks D1 and D2, of radii r1 and r2 and cen-troidal moments of inertia I1 and I2, are hingedat points O1 and O2, respectively. Rolling withoutslipping is enforced at the contact point. A blockAof mass m is attached to an inextensible, masslessstring that wraps around D2, and can move onlyin the vertical direction. Gravity g is acting down-wards, as shown. A constant torque M0 is appliedto D1. What is the magnitude a of the accelerationof A?

(a) a =M0

r2+mg

m+ I1r22

+ I2r21

(b)I a =M0

r1−mg

m+ I1r21

+ I2r22

(c) a =M0

r1−mgm

(d) a = 0

(e) a =

M0

r21−mg

m+ I1r1

+ I2r2

M3m 3m

kk

3M↵k↵k

kk

x1 x2 x3

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kk

11. The two systems shown in the figure feature asymmetric vibration mode U = [−1 0 1]. Themasses and the elastic constants of the springsare indicated in the drawing. What is the valueof α that guarantees that the eigenfrequency ofthis mode is the same for the two systems?

(a)I α = 1

(b) α = 3Mm

(c) α = 3

(d) α =√3Mm

(e) α =√

3

9

12. Consider an arbitrary 3D rigid body of uniform density ρ. Which of these state-ments is not true?

(a)I Two principal axis can be parallel to each other in case the body has anappropriate symmetry.

(b) The body has three principal axes of inertia.

(c) The three principal moments of inertia can be equal.

(d) The moment of inertia tensor of the body is proportional to ρ.

(e) The center of mass can lay outside the body.

m

m

mm

m

m

Smooth circular guide

k1

k2k3

k4

k5

k6

k7

13. The planar system depicted in the figure con-sists of six particles of equal mass m con-nected by linear springs of stiffness ki, withi = 1, . . . , 7 and zero unstretched length, asshown. Four particles slide on a frictionless cir-cular rail, while the two internal particles arefree to move in any direction. No gravity is act-ing on the system. The particles are given anarbitrary, non-equilibrium initial position andthen released from rest to set the system in mo-tion. Which of the following statements mustbe true?

(a)I The work done by the internal forces is not zero.

(b) The work done on the masses sliding on the rails is zero.

(c) The angular momentum of each particle about an arbitrary point is con-served.

(d) The linear momentum of the system is not conserved.

(e) The linear momentum of a particle on the rail is conserved.

10

x

y

↵

v0

O

trajectory

g

B

m

ground

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14. A particle of mass m is launched from point Owith velocity v0 at an angle α. It hits point Bwith coordinates xB = 3L, yB = L, as shown.What is the correct expression for v0 = |v0|?

(a)I v0 =√

92

gLcos2 α

13 tanα−1

(b) v0 =√

92

gLtanα

13 tanα−2

(c) v0 =√

12

gcos2 α

12 tanα−1

(d) v0 =√

112

gLcos2 α

13 tanα

(e) v0 =√

92

gLsin2 α

1tanα−1

v0

O

trajectory m

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15. A particle of mass m moves on a horizontal,frictionless plane, launched from point A withcoordinates (x, y) = (0, r0) and with horizontalvelocity v0 = v0ex. The particle is subject to aforce F = mk2rer, where k is a constant, r isthe distance of the particle from point O, ander is a radial unit vector. Which is the cor-rect equation describing the trajectory of theparticle?

(a)(xv0

)+(yr0

)= 1

(b)(xv0

)2+(

ykr0

)2= 1

(c)I(

xv0/k

)2+(yr0

)2= 1

(d)(xv0

)−(

ykr0

)= 1

(e)(

xv0/k

)2+(yr0

)2= r20

11

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16. A particle of mass m and of negligible dimen-sions is impacting a rigid wall at an angleα = π/4 with velocity v. What is the maxi-mum range for the coefficient of restitution ethat allows the particle to enter the slot shownin the figure after impact? Ignore the effect ofgravity and let β = π/6.

(a)√32< e < 1

(b) 1√2< e < 1

2

(c) e < 1

(d) 0 < e < 1

(e)I 1√3< e < 1

17. A particle of mass m is connected to a spring ofstiffness k and unstretched length L0. It movesin a pipe, which rotates around point O witha prescribed angular velocity Ω = Ω(t)e3. Noforces are applied to the particle. Which of thefollowing statements is true?

(a) The centrifugal force is zero when thespring is unstretched.

(b) The Coriolis force is always zero.

(c)I The Coriolis force is acting perpendicularto the pipe.

(d) The centrifugal force is zero when the spring reaches its maximum stretch.

(e) The Coriolis force is always directed along e3.

12

g F

M

m

L

cord

Frictionless surface

A

B

↵

C18. A cart of mass M is towed by a constant force

F on a frictionless surface. A uniform bar ofmass m and length L is hinged onto the cartat point A, and held at a constant inclinationangle α by an inexstensible cord connectingpoints B and C. In the configuration shown,the chord is horizontal. Gravity acts down-wards, as shown. The tension in the cord isdenoted by T . What is the magnitude F ofthe towing force?

(a) F = m+Mm

(T −mg cosα)

(b) F = Mm

(2T −mg sinα)

(c)I F = m+Mm

(2T −mg tanα)

(d) F = 0

(e) F = 2T −mg tanα

gm

L

torsional spring

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19. A bar of length L and mass m rotates aboutits tip O in a vertical plane. Gravity is act-ing downward, as shown. A torsional spring ofelastic costant k = mgL

π2 is acting on the barand provides a restoring moment M = −kθ.When θ = π/2, the bar has an angular veloc-ity of Ω. What is the condition on Ω for thebar to reach θ = π? The centroidal moment ofinertia I of the bar is given by I = 1

12mL2.

(a) Ω = −θ

(b) Ω >√

32gL

(c) Ω <√

57gL

(d)I Ω >√

214gL

(e) Ω > 0

13

20. Consider a rigid body moving in a three dimensional space, subject to an arbitraryresultant force and resultant torque. Consider then the Euler equations in theform

I1ω1 + (I3 − I2)ω2ω3 = M1,

I2ω2 + (I1 − I3)ω1ω3 = M2,

I3ω3 + (I2 − I1)ω1ω2 = M3.

Which of the following statements is true?

(a)I The Euler equations of motions are valid only with respect to the principalframe of reference.

(b) The Euler equations of motion can be derived from the angular momentumprinciple for the rigid body, with respect to any reference frame attached tothe body.

(c) The Euler equations of motion govern the trajectory of the center of massof the rigid body.

(d) The Euler equations of motion describe the motion of the rigid body withrespect to any inertial frame of reference.

(e) The Euler equations of motions can be derived from the angular momentumprinciple for the rigid body, with respect to any fixed point outside the body.

14

Solution Version n. 1

1. c

2. e

3. d

4. b

5. c

6. e

7. b

8. c

9. b

10. a

11. a

12. a

13. a

14. c

15. e

16. c

17. c

18. d

19. a

Solutions – Page 1

dynamics - retake exam 27.08.2018- type 1 - n.ethz.chkiranku/dynamics2018/files/dynamics-final exam...

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