Practice Exam Dynamics

October 7, 2015

1 Question

A projectile is launched with speed v0 from point A. Deter-mine the launch angle which results in the maximum range Rup the incline of angle α (where 0 6 α 6 90).

(1) θ = 90+α2 (deg.)

(2) θ = 90−α2 (deg.)

(3) θ = 45− α (deg.)(4) θ = 45 + α (deg.)

2 Question

A particle of mass m is attached to point O by an inextensiblestring of length l. Initially the string is slack when m is movingto the left with a speed v0 in the position shown. Calculate thespeed of m just after the string becomes taut.

(1) v = v0(2) v = d2

l2 v0(3) v = d

l v0(4) v = c

l v0

3 Question

Each of the sliders A and B has a mass of 2 kg and moveswith negligible friction in its respective guide, with y being inthe vertical direction. A horizontal force of 20 N is applied tothe midpoint of the connecting link of negligible mass, and theassembly is released from rest with θ = 0. Calculate the veloc-ity vA with which A strikes the horizontal guide when θ = 90.Assume that the acceleration due to gravity, g, equals 10 m/s2.Note that the slider A can only move in the vertical guide.

(1) vA =√

12 m/s(2) vA = 2 m/s(3) vA = 2

√2 m/s

(4) None of the above

1

4 Question

The figure shows n spheres of equal mass m suspended in a lineby wires of equal length so that the spheres are almost touchingeach other. If sphere 1 is released from the dashed position andstrikes sphere 2 with a velocity v1, write an expression for thevelocity vn of the nth sphere immediately after being struck bythe one adjacent to it. The common coefficient of restitutionis e.

(1) vn = ( 1+e2 )n−1v1

(2) vn = ( 1−e2 )n−1v1

(3) vn = ( 1+e4 )n−1v1

(4) vn = ( 1−e4 )n−1v1

5 Question

The simple pendulum A of mass mA and length l is sus-pended from the trolley B of mass mB . The friction is neg-ligible in the system. If the system is released from restat θ = 0, determine the velocity vB of the trolley whenθ = 90.

(1) vB = mA

mB

√2gl

1+mAmB

(2) vB = mB

mA

√2gl

(3) vB = mB

mA

√2gl

1+mBmA

(4) vB = mA

mB

√2gl

2

6 Question

A mass m is on an inclined surface at angle θ, as shown in the picture.Attached to the mass is a spring of constant k that is fixed at the topof the inclined surface. Consider gravity in the problem and µk as thedynamic friction constant. Find the equation of motion in terms of thevertical coordinate x assuming that the unstretched length of the springis l0.

(1) mxsin θ + ( x

sin θ − l0)k + xsin θµk = mg sin θ

(2) mxcosθ + ( x

cosθ − l0)k + xcosθµk = mg sin θ

(3) mxsin θ + ( x

sin θ + l0)k + xsin θµk = mg sin θ

(4) None of the above

7 Question

A stationary mass mA is viewed by an observer P who is sit-ting on a small merry-go-round which rotates about a fixedvertical axis at B with a constant angular velocity Ω as shown.Determine the magnitude of the Coriolis force associated withthe mass mA in the frame of the observer P . Does this forcedepend on the |rBP | distance?

(1) 2mA|rBP |R Ωd; depends on the location of the observer.

(2) 2mAΩ2d; does not depend on the location of the observer.(3) 0; does not depend on the location of the observer.(4) None of the above

8 Question

A wheel of radius r rolls without slipping on a concave circularsurface of radius R with angular velocity ω relative to the sur-face. The position of center C of this wheel can be describedby the angle ρ, as shown. Determine the angular velocity ofthe wheel.

(1) (R−r)ωR

(2) Rωr

(3) ρ(4) ω

3

9 Question

A drum of radius b turns clockwise at a constant angular ve-locity ω and causes the carriage P to move to the right as theunwound length of the connecting cable is shortened. Derivean expression for the velocity v of P in the horizontal guide interms of the angle θ.

(1) v = bωsinθ

(2) v = bωsinθ(3) v = bω cos θ(4) None of the above

10 Question

Which free body diagram properly represents correctly the sys-tem shown in the figure?

(1) a(2) b(3) c(4) d

4

11 Question

Ball 2 is originally at rest. Ball 1 strikes ball 2 with an initialvelocity of v1 with an angle φ 6= 0, as shown. The velocity ofthe two balls after the impact will be u1 and u2 respectively.If both balls have the same mass and the collision is perfectlyelastic, determine the angle θ after collision. Neglect the sizeof each ball.

(1) θ = 90

(2) θ = 45

(3) θ = 30

(4) The angle φ is required to answer this question.

12 Question

A small ball of mass m is given an initial velocity vA atA parallel to the horizontal rim of a smooth bowl. LetC(t) denote the projection of point A along the z axis (i.e.rC(t) = h(t)k). Which one of the following sentences is incor-rect?

(1) The angular momentum about any fixed point of the z axis is con-served.(2) The angular momentum about point C(t) is conserved.(3) The energy is conserved.(4) The active forces and the reaction forces do not contribute to the re-sulting torque about the z axis.

5

13 Question

A bob of mass m, attached to the rope, rotates along the horizontal cir-cular path with constant angular velocity ω. The tension in the rope isdenoted by T . Which of the following is the correct free body diagram ofactive forces acting on the bob?

(1) a(2) b(3) c(4) d

6

14 Question

Consider a pendulum consisting of a spring and a damper with a massm on the end, as shown in the figure. Let r = |rOP | denote thelength of the pendulum; the spring and damping coefficients are kand c, respectively. The length of the unstretched spring is l; θ de-notes the angle between the vertical line and the pendulum. Assum-ing the motion to be planar, find the equations of motion of the sys-tem.

(1) mr −mrθ2 −mg cos θ + k(r − l) + cr = 0

mr2θ + 2mrrθ +mgr sin θ = 0

(2) mr +mrθ2 −mg cos θ + k(r − l) + cr = 0

mr2θ − 2mrrθ +mgr cos θ = 0

(3) mr −mrθ2 −mg + k(r − l) + cr = 0

mr2θ − 2mrrθ −mgr = 0

(4) mrθ2 +mg cos θ + k(r − l) + cr = 0

2rθ − g sin θ = 0

15 Question

A wheel of radius r is free to rotate about the bent axleCO which turns about the vertical axis at a constant ratep [rad/s]. If the wheel rolls without slipping along acircular track of radius R, determine the expressions forthe angular velocity ω and angular acceleration α of thewheel in the moving (x, y, z) frame. (The x-axis is al-ways horizontal, and the y-axis remains orthogonal to x andz.)

(1) ω = p[f

2cos θ + f

3(sin θ +R/r)]

α = (Rp3/r cos θ)f1

(2) ω = p[f

2cos θ − f

3(sin θ +R/r)]

α = (Rp3/r sin θ)f1

(3) ω = p[f

2sin θ + f

3(cos θ +R/r)]

α = (Rp3/r cos θ)f1

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(4) ω = p[f

2cos θ + f

3(cos θ +R/r)]

α = (2Rp3/3r cos θ)f1

16 Question

A thin disk of mass m1 spins about its axis eψ with

a constant angular velocity ψ as shown in the figure.The block at B has mass m2. By adjusting its posi-tion s, it is possible to change the precession speedof the disk about eϕ while the shaft remains hor-izontal. Determine the position s that will enablethe disk to have a constant precession rate ϕ aboutthe pivot. Neglect the weight of the shaft.Hint: The moment of inertia of the disk with respectto the eψ axis is Iψψ = 0.5m1r

2.

(1) s = 2gm1l+ψϕr2m1

2m2g

(2) s = 2gm2l−ψϕr2m1

2m1g

(3) s = m1

m2l

(4) None of the above.

17 Question

A disk of mass m and radius r rolls without slipping on a cir-cular surface of radius R. Determine the maximum angular ve-locity ωmax of the disk as it crosses the vertical, assuming thatit has been released with zero velocity from θ(0) = θ0. Ifthe motion is confined to small vibrations (i.e., θ ≈ 0), de-termine the natural frequency ω0 of these vibrations. (Cau-tion: Do not confuse the disk’s angular velocity with θ or withω0.)

(1) ωmax =

2

r

√(R− r)(1− cos θ0)g

3

ω0 =

√2g

3(R− r)

(2) ωmax =

1

r

√R(1− cos θ0)g

3

ω0 =

√4g

3(R− r)

8

(3) ωmax =

R

rω0θ0

ω0 =

√3g

2(R− r)

(4) ωmax =

√R(1− cos θ0)g

3

ω0 =

√3g

2(R− r)

18 Question

A rigid body with rotational symmetry rotates at a constant angular velocity ωabout its axis of symmetry z, which is fixed. Denote the angular momentumwith respect to a fixed point O on the z-axis by HO. Which statement is cor-rect?

(1) HO is constant.(2) |HO| is constant but HO/|HO| is not.(3) HO/|HO| is constant but HO is not.(4) None of the above.

19 Question

Two uniform cylinders of mass m1 and m2 and radius R1 and R2 arewelded together. This composite object rotates without friction about afixed point O. Two inextensible massless strings are wrapped withoutslipping around the larger cylinder. The two ends of the strings are con-nected to the ground by a spring of constant k and a damper of constantc. The smaller cylinder is connected to a block of mass m0 via an in-extensible massless string wrapped without slipping around the smallercylinder. The block is constrained to move only vertically. Determine thevalues of c for which the system performs underdamped vibrations.Hint: The moment of inertia of the disk at its centroid relative to thez-axis, which is perpendicular to the plane, is Izz = mr2

2 .

(1)

c <2

√kR2

2((m0 + m1

2

)R2

1 +m2R2

2

2 )

R22

.

(2)

c <=2

√kR2

2((m0 + m1

2

)R2

1 +m2R2

2

2 )

R22

.

9

(3)

c >2

√kR2

2((m0 + m1

2

)R2

1 +m2R2

2

2 )

R22

.

(4)

c <

√kR2

2((m0 + m1

2

)R2

1 +m2R2

2

2 )

R22

.

20 Question

Two cylinders of the same size and mass roll downan incline, starting from rest. Cylinder A has mostof its mass concentrated at the rim, while cylinderB has most of its mass concentrated at the center.Which reaches the bottom first?

(1) Cylinder A(2) Cylinder B(3) Both at the same time.(4) Depends on the static friction coefficient.

21 Question

Two disks at rest of radiusR are separated by a spin-dle of smaller diameter. A string is wound aroundthe spindle and pulled gently by the force F indi-cated in the figure, bringing the disk pair into rollingmotion. At what angle θ does the direction of rollingchange ?

(1) θ = cos−1( rR )(2) θ = 0

(3) θ = 90

(4) None of the above.

22 Question

Derive the natural frequency ω0 of the system shownfor the case k1 = k2 = k and m1 = m2 = m.Assume that the mass of the linkage is negligi-ble.

(1) ω0 =√

km

(2) ω0 =√

abkm

(3) ω0 =√

bakm

(4) None of the above.

10

23 Question

Calculate the natural frequency ω0 of the system shownin the figure. The mass and the friction on the pulleysare negligible.

(1) ω0 =√

k5m

(2) ω0 =√

k2m

(3) ω0 =√

k3m

(4) None of the above.

24 Question

The two identical steel frames with the dimensions shown are fab-ricated from the same bar stock and are hinged at the midpoints Aand B of their sides. If the frame is resting in the position shown ona horizontal surface with negligible friction, determine the velocityv with which the upper ends of the frame hit the horizontal surfaceif the cord at C is cut.

Hint:

(1) v =√

12gb c+2b3c+4b cos θ2

(2) v =√

8gb c+2b3c+4b cos θ

(3) v =√

8gb b+c3c+b cos θ

(4) v =√

12gb b+c3c+b cos θ2

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25 Question

Determine the highest natural frequency ω andthe corresponding mode shape u for the twodegree-of-freedom system shown in figure. As-sume the state vector is defined as x =[x1, x2]T .

(1) ω =√

11k2m ; u = [1,−0.5]T

(2) ω =√

km ; u = [1, 1]T

(3) ω =√

3km ; u = [1,−1]T

(4) ω =√

5k2m ; u = [1,−0.5]T

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