ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 1 Given: Block A (having a mass of m) is attached to cart B with two springs of

stiffnesses 3k and k, as shown below. A third spring of stiffness 2k is attached between A and ground. Cart B is given a PRESCRIBED DISPLACEMENT of xB(t) = b cosωt. The absolute motion of block A is described by the coordinate x. All springs are unstretched when x = xB = 0. Consider all surfaces to be smooth.

Find: For this problem:

a) Derive the differential equation of motion (EOM) for block A using the coordinate x(t). NOTE: Clearly indicate the four solution steps in your work.

b) Determine the numerical value for the natural frequency for this system. Use m = 12 kg and K = 800 N/m.

c) The frequency of the base motion is ω = 25 rad/sec with the amplitude of the motion of the cart given by b = 0.1 meters. Determine the particular solution xP(t) of the EOM for this problem. You need to show the derivation of the form of the particular solution. Do NOT write down the solution form from memory.

k

x+

3k

2k

xB+

m

A

B

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 2 Given: A thin, homogeneous bar of length 3 meters and mass m = 200 kg is

constrained to move on a smooth HOIZONTAL plane. In addition, ends A and B are constrained move in smooth slots aligned with the y- and x-axes, respectively. A CCW torque M = 300 N-m is applied at the midpoint of the bar. The bar is released from rest with θ = 36.87°.

Find: Determine the angular acceleration of the bar when the bar is released from

rest. Write your answer as a vector. NOTE: Clearly indicate the four solution steps in your work.

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 3 Given: Particles A and B each have a mass of m =10 kg. A is constrained to move on

a horizontal arm OC, and B is constrained to move on the vertical shaft about which arm OC rotates. A single, taut cable connects A and B as shown in the figure below. At an instant when the shaft is rotating with a rate of ω1 = 30 rad/sec and R = 0.2 meters, A and B are released from rest relative to the rotating arm and shaft.

Consider all surfaces to be smooth and assume the masses of the arm, shaft and pulleys to be negligible.

Find: Find the speed of B when A has moved an additional 0.3 meters OUTWARD

on arm OC. NOTE: Clearly indicate all solution steps in your work. A

B

O

R

g

ω1

C

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 4 Part a) - 2 points Particles A and B are attached to a rigid rod with the rod being pinned to ground at point O. A bullet “b” strikes particle A and sticks. Consider a system made up of b, A, B and the rod. Circle ALL answers below that correctly describe this system during the impact of b with A:

a) linear momentum is conserved.

b) angular momentum about point A is conserved. c) angular momentum about point O is conserved.

d) energy is conserved. e) none of the above.

Part b) - 2 points Particle A (having a mass of m) is traveling to the left with a speed of vA1 when it strikes a stationary particle B (having a mass of 2m). The coefficient of restitution for the impact of A and B is known to be e = 0. Circle the answer below which most accurately describes the speed

vA2 of A immediately after it impacts B.

a)

vA2 = 0

b)

vA2 = vA1 /3

c)

vA2 = vA1 /2

d)

vA2 = 2vA1 /3

e)

vA2 = vA1

f)

vA2 > vA1

m

VB1 = 0

2m A B

VA1

smooth

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 4 (continued) Part c) - 2 points Shown below left is a thin ring (of mass m, outer radius R and center A) that is released from rest on a rough incline. Shown below right is a homogeneous disk (of mass m, outer radius R and center B) that is released from rest at the same height of the ring on rough incline. Both the ring and disk roll without slipping on their respective inclines. Let

vA2 and

vB2

represent the speeds of A and B, respectively, after both A and B have dropped through vertical distance of H. Circle the answer below that most accurately describes the relative sizes of

vA2 and

vB2

:

a)

vA2 > vB2

b)

vA2 = vB2

c)

vA2 < vB2

θ no slip

R

A

A

m

θ no slip

R

B

B

m

H

vB 2

vA2

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 4 (continued) Part d) - 2 points Aircraft A is traveling along a straight path with a speed of

vA . Aircraft B is traveling along a circular path of radius R with a speed of

vB

. Circle the answer below that most accurately represents the observed velocities of A and B:

a)

vA /B = vA ! vB is the velocity of aircraft A as seen by the pilot of aircraft B.

b)

vB /A = vB ! vA is the velocity of aircraft B as seen by the pilot of aircraft A.

c) Both a) and b).

d) Neither a) nor b). HINT: Consider the general form of the moving reference frame velocity equation and

the motion of the observer in each case.

vB

vA d

R

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 4 (continued) Part e) - 2 points A force F acts at the end of a cable that has been wrapped around the inner surface of a stepped drum, as shown in the figure drawn to scale below. The friction force between the outer surface of the drum and the ground is sufficient to prevent slippage between the drum and ground. Circle the answer below that most accurately describes the initial motion of the drum:

a) The drum moves to the right. b) The drum moves to the left. c) The drum cannot move without slipping.

Part f) - 2 points Circle ALL situations below for which the “short form” of Euler’s equation

MA! = IA" is valid for analyzing the planar motion of a rigid body:

a) Point A is a non-accelerating point. b) Point A is the center of mass G.

c) The acceleration of A is parallel to the acceleration of the center of mass G. d) The vector

rG /A is parallel to the acceleration of point A where G is the center of mass.

e) A can be ANY point on the rigid body.

no slip

F

O

G

A

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 4 (continued) Part g) - 2 points State in words Newton’s THIRD law of motion.

Part h) - 2 points A force F acts on the center of a wheel as the wheel rolls without slipping along a rough horizontal surface. Explain in words why friction does not do work on the wheel as it rolls.

O

no slip

F

vO

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 5 Part a) - 5 points A cannonball P of mass m is fired toward a steel barrier on a stationary cart. At some time after rebounding from the barrier, the cannonball is observed to have a speed of vP = 30 ft/sec and moving in a direction shown below in the figure. Let M be the combined mass of the cannon and cart. If m/M = 0.1, what is the velocity (both magnitude AND direction) of the cart after the cannonball bounces off the steel barrier? Assume that the cart moves without friction along a horizontal surface and ignore the influence of air resistance.

20°

vP P

cannon

cart

x

y

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 5 (continued) Part b) - 5 points The homogeneous disk shown below has a mass of m = 16 kg and outer radius of R = 0.5 meters

• The disk is gently placed on a rough horizontal surface with a speed of

vO1 = 20 meters /sec to the left and no angular velocity (

!1 = 0) at position 1. • The disk initially moves to the left with slipping (with both translation and

rotation) until reaching position 2. • At position 2 the slipping ceases and it begins to roll without slipping. At position

2 it is known that

vO2 =10 meters /sec . Determine the work done by friction on the disk as it moves from position 1 to position 2. HINT: Consider the work-energy equation for the disk. You are given information from

which you can compute the initial and final kinetic energies of the disk.

SLIP

R

O

vO1

!1 = 0

O

vO2

!2

NO SLIP

position 1 position 2

rough

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 5 (continued) Part c) - 5 points Particle P (having a mass of m = 4 kg) is traveling with a velocity of

v = 15i + 20 j( ) m /sec when a net force of

F = 100 j + 280k( ) newtons acts on P.

a) At this instant, determine the rate of change of speed of P. b) Is the speed of P increasing or decreasing at this instant? Provide justification for

your answer.

x

y

z

v

F

P

ME 274 – Fall 2007 Name Final Examination PROBLEM NO. 5 (continued) Part d) - 5 points The time history x(t) for the free response of the undamped spring-mass system is shown below where xst = 0.05 meters. Determine the natural period of free response τ for this system. HINT: Use the relationship between k/m and the static deformation xst in solving this problem.

k

x+ m

g

x(t)

t τ

xst