BME 2220 Homework 2 _2_
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BME 2220 Homework 2
Due February 17, 2022
Problem 1
During a match, a wrestler pushes on the head of his opponent with force
F=1200 N, as shown in the picture on the left. His opponent holds his head
in equilibrium by applying an isometric contraction force in his neck muscle
(M). The musculature in the neck is very complex anatomically, but for the
sake of this problem, assume the neck muscles can be simulated as a single
cable. Treat the connection of the head to the C1 vertebrae as a hinge
(point C). By chance, the lines of action of F, M, and Cy all happen to pass
through point A in this problem. (Note that the lack of dimensions in this
problem is intentional. They aren’t needed to solve the problem.)
Using the free-body diagram provided on the right, determine the
magnitude of the force in the neck muscles (M), as well as the reaction
forces at the neck joint (Cx and Cy), needed for the opponent to hold his head in equilibrium. (Units: N)
Problem 2
Suppose you have been asked to find the Young’s modulus of an unknown material. The geometry of the material
sample is shown below. It has an original length L = 8 cm, and a perfectly square hollow cross-section, with
dimensions shown on the right. When you apply a force F = 150 N to the sample as shown below, the sample
deforms to a new length L
new
=8.10 cm.
a.
Calculate Young’s Modulus (units: N/cm2).
b.
Calculate the normal and shear stress on the angled plane drawn in the figure.
Ɵ =
65°. Ignore weight
(units: N/cm
2
)
F = 1200 N
65°
F = 150 N
F = 150 N
cut
Ɵ
EMA
D
Ncos
Gs
X
FUSCO
X
Mcoscos
Fios
30
m
t
Fx
Fios
30
M
cos
65
Cx
D
Cx
0
spy
Cy
Fs
ing
o
t
Msm 65
12008
to
t
2459.029
s
co
Cy
2828
627
N
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2.4826
2 150
N
as
54.75
2
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2
vs
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Problem 3
A frame with negligible mass is supported by a hinge at A and a rope (cable
connections at “B” and “D”) that passes over a smooth pulley at C, as shown to the
right. Determine the force in the cable and the reaction force(s) at A (units: N). Your
F.B.D. will be of the frame.
Problem 4
Given the beam below:
a.
Draw a free body diagram for the beam. Ignore its weight.
b.
Calculate the reaction forces at A and B (units: N)
c.
Determine the normal and shear forces (units: N) and the internal moment (units: N
∙
m) along the cut.
Problem 5
A beam with a length of 1.2 m and negligible weight is firmly attached
as a fixed support on the left end. A mass of 4 kg, attached at x=0.6 m,
pulls down to create force W. A cable pulls up with force P of 100N at
x=0.2m, and a force R of 30N is applied to the end of the beam.
a.
Draw a FBD of the beam.
b.
Calculate the values of the reaction forces (units: N) and
moment (units: Nm) generated at the fixed support.
c.
Calculate the internal forces and moment at the point A.
0.6 m
0.6 m
0.4 m
0.2 m
3 m
1 m
40°
900 N
Tae Tae
T
EM
a
TC
3
90061
TaosCho
TsinCud
Tf
Tnc
E
Fx
900
t
TO
CYO
t
Fax
O
E
it
y
Ts
i
n
no
T
t
Fay
o
1
SE
scyoj
s.na
9871.8910
FAX
900
Tco
Cyo
8
462.307N
Fay
Tsin
Cao
T
16217.420
N
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Af
Agt
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B
m
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0
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933.332
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N
Normal
Forces
Goo
o
i
n
Nx
shear
Fees
566ait
N
Ny
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R 30N
70
C
a
A
MS
ooh
oom
o om
Tx
Pcos
70
R
cos
oo
o
Fx
0
Tx
Pcos
7
offresco
100
cacao
t
3
cases
49.202N
EFy
0
Ty
Psin
70
Rin
Ga
Wg
0
Ty
Rsin
oo
Ps
n
7
tug
2
788
N
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7070.2
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392
Nm
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Problem 6
While relaxing in a hammock on vacation, Jane starts to think about biomechanics. If Jane weighs
55kg, determine the tension on the two support ropes as shown to the right (units: N). Angle A is
equal to 45° and angle B is equal to 35°. Your F.B.D. will be of Jane connected to the tree by cable
connections (i.e. the ropes). Ignore the weight of the hammock and ropes.
Problem 7
You are performing a uniaxial compression test on a specimen containing a bone and a solid
metal graft. The bone has two segments, a spongy, trabecular segment and a hollow,
cortical segment (as shown at right). The outer section of the hollow bone and the entire
cross section of the spongy bone can be treated as solid materials. Ignore the weight of the
specimen in your calculations. The hollow bone has an inner radius of 0.5 cm. The graft and
the bone have an outer radius of 1 cm. Before loading, each segment is 2 cm long. Suppose
the hollow bone has a Young’s modulus (E) of 10 GPa, and the spongy bone has a Young’s
modulus of 1 GPa. You load the bone with a 55
N force. After loading, the entire specimen’s
length is 5.94 cm.
a.
Calculate the normal stresses in the hollow bone, spongy bone, and graft (units:
MPa).
b.
Calculate the change in length in each of the three segments (units: mm).
c.
Calculate the Young’s modulus of the graft (units: MPa)
Problem 8
After the compression test in problem 7, Rebecca unloads the specimen and it returns to its original dimensions.
Now she applies an 80 N
∙
m torque to the top of the specimen, which is firmly fixed to the surface at its base.
a.
Calculate the shear stress at the inner and outer radius of the hollow bone (units: N/cm2).
b.
With the same torque in place, calculate the maximum shear stress in the solid metal graft (units: N/cm2).
F = 55 N
44
w
55 9.8
1
5
I
539N
NW
Fx
A
cos
us
B
cos
635
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cos
451
13801135
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35
W
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5
539
Bws9
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539
B
1392728
1
539
13
387.0180225
A
429179
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448.3433
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a
Sponge
I
man
random
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0.175
MPa
0.2334
MPa
0.175
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5.94
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0.175
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1
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3.501
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0.596
mm
c
E
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0 175120
5.8726hPa
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a
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- Don't Use Chat GPT Will Upvote And Give Handwritten Solution Pleasearrow_forwardaccount_circle Science PhysicsQ&A LibraryA child’s toy consists of a m = 31 g monkey suspended from a spring of negligible mass and spring constant k. When the toy monkey is first hung on the spring and the system reaches equilibrium, the spring has stretched a distance of x = 17.6 cm, as shown in the diagram. This toy is so adorable you pull the monkey down an additional d = 7.6 cm from equilibrium and release it from rest, and smile with delight as it bounces playfully up and down. 1. Calculate the speed of the monkey, ve, in meters per second, as it passes through equilibrium. 2. Derive an expression for the total mechanical energy of the system as the monkey reaches the top of the motion, Etop, in terms of m, x, d, k, the maximum height above the bottom of the motion, hmax, and the variables available in the palette. 3. Calculate the maximum displacement, h, in centimeters, above the equilibrium position, that the monkey reaches. A child’s toy consists of a…arrow_forward1. The PASCO human arm model is configured such that the cord representing the bicep is perfectly vertical and the forearm is at 90° (in the figure to the right, the cord is not quite vertical). A mass of 100 g is attached to the hand. Draw a free-body diagram on the figure to the right showing all forces which act on the forearm. The force of the bicep F on the arm The force of the humerus FH on the arm The weight of the forearm W The mass in the hand Wm 100 g Be careful to draw the force vectors with tails beginning at the point where the force is actually applied to the forearm. 2. Consider the free body diagram below. Determine the perpendicular component F̟ of the force F exerted by the biceps brachii on the forearm. Use the fact that cos 0 = H/B to write this component directly in terms of the humerus length H and the biceps length B. H 3. If the forearm is in equilibrium, then there is no angular acceleration and therefore the sum of the torques applied to the forearm must be…arrow_forward
- The figure shows a person wearing weight boots and doing lower leg flexion/extension exercise in a sitting position to strengthen the quadriceps muscles and a simple mechanical model of his leg. W1 is the weight of the lower leg, W0 is the weight of the boot, the magnitude of the pulling force applied to the tibia by the quadriceps muscles through the FM patellar tendon, the magnitude of the reaction force acting on the FJ tibiofemoral joint. Point O is the center of the tibiofemoral joint, point A is the point where the patellar tendon attaches to the tibia, point B is the center of gravity of the lower leg, point C is the center of gravity of the weight boot. The distances between point O and points A, B and C were measured as a=13 cm, b=27 cm and c=36 cm, respectively. The angle that the long axis of the tibia makes with the horizontal is β=34°, the angle between the line of action of the quadriceps muscle strength and the long axis of the tibia is α=18°. Points O, A, B and C lie…arrow_forwardNeeds Complete typed solution with 100 % accuracy.arrow_forward3. A uniform, 200-kg plate is kept in static equilibrium by two hinges, A and B, and a single cable, DCE. a) Draw the FBD of the plate. b) Write the coordinates of each named point in the figure (A, B, etc.) and any additional point where a force or moment is acting. c) Is this system statically determinate? Explain your answer. d) Is this system completely (i.e., properly) constrained? Explain your answer. e) Assume the hinges are properly aligned and do not exert any reaction moments on the plate. Determine the tension force on the cable DCE for this plate to stay in static equilibrium. (Hint: when a rigid body is in static equilibrium, the resultant moment about any point or line has to be equal to zero. Could you find a line that gets rid of your other unknowns?) 50 cm D 90 mm 90 cm с 30 cm 120 cm B 90 mm E. 80 cm xarrow_forward
- attached is a past paper question in which we werent given the solution. a solution with clear steps and justification would be massively appreciated thankyou.arrow_forwardPhysics 121 Spring 2021 - Document #11: Homework #04 & Reading Assignment page 4 of 8 Problem 1: Gnome Ride - This from a Previous Exam I. A Gnome of given mass M goes on the Gnome Ride as follows: He stands on a horizontal platform that is connected to a large piston so that the platform is driven vertically with a position as a function of time according to the following equation: y(t) = C cos(wt) Here w is a constant given angular frequency, C is a given constant (with appropriate physical units) and y represents the vertical position, positive upward as indicated. Part (a) - What is the velocity of the Gnome at time t = 0? Explain your work. Present your answer in terms of the given parameters Part (b) – What is the net force on the Gnome at time t = 0? Explain your work. Present your answer in terms of the given parameters Part (c) – What is the Normal Force on the Gnome at time t = 0? Explain your work. Present your answer in terms of the given parameters Some Possibly Useful…arrow_forwardProblem 7 Marooned You are stranded on a boat with no way to paddle to the shore (and no water current to move the boat). You have mass 2m and the boat also has mass 2m. In addition, you brought 2 weights from the physics lab that each have mass m. Since you have been hitting the gym with Juan, you can throw the weights with an impressive speed of vo. (a) If you throw both weights off the back of the boat simultaneously, what will the speed of the boat (and you) be? (b) Suppose instead you throw each weight off the back of the boat individually. What will the speed of the boat be in this case? (Be careful when considering how fast the second weight is thrown with respect to the water). (c) If you were trying to get to shore as fast as possible, which method should you choose? Should you throw the weights towards shore or away from shore?arrow_forward
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