Worksheet (Assignment)
1.As illustrated in Fig. below , consider the person performing extension/flexion
movements of the lower leg about the knee joint (point O) to investigate the forces
and torques produced by muscles crossing the knee joint. The geometric
parameters of the model under investigation, some of the forces acting on the lower
leg and its free-body diagrams are shown in Figs. below. For this system, the
angular displacement, angular velocity, and angular acceleration of the lower leg
were computed using data obtained during the experiment such that at an instant
when θ = 65_, ω = 4.5 rad/s, and α = 180 rad/s2. Furthermore, for this system
assume that a = 4.0 cm, b = 23 cm, β = 25_, and the net torque generated about
the knee joint is M0 =55 Nm. If the torque generated about the knee joint by the
weight of the lower
leg is Mw =11.5 Nm, determine:
(a) The mass moment of inertia of the lower leg about the knee joint
(b) The weight of the lower leg
(c) The tension in the patellar tendon
(d) The reaction force at the knee joint
2.Consider a person performing shoulder exercises by using a dumbbell. The forces
acting on the arm and the mechanical model of the system are shown in Fig. below .
For this system assume that the arm of the person is fully extended to the
horizontal. Point O designates the axis of rotation of the shoulder joint, A is the point
of attachment of the deltoid muscle to the humerus, point B is the center of gravity
of the entire arm, and point C is the center of gravity of the dumbbell. The distances
between the axis of rotation of the shoulder joint (point O) and points A, B, and C
are a = 17 cm, b =33 cm, and c =63 cm. The dumbbell weighs W0 = 64 N and for
this position of the arm it is estimated that the magnitude of the muscle force is FM
=1051 N. If the lines of action of the muscle (FM) and the joint reaction forces (FJ)
make an angle θ = 18_ and β=12_ with the horizontal, respectively:
(a) Determine the magnitude of reaction force (FJ) at the shoulder joint.
(b) Determine the total weight (W) of the arm.
(c) Determine the magnitude of the muscle (FM1) and joint reaction (FJ1) forces when
the weight of the
dumbbell is increased by 5 N
3. A tooth is both moved and rotated by the application of the two forces indicated
in the figure at right. Set the torque about the center of the tooth equal to 0.0099
N·m and the sum of the forces equal to 1.8 N in order to determine the magnitudes
of the forces. Now rearrange and solve for F1 and Solve for F2=F total-F1.If D=0.0045m
and d=0.0032.
4. The masseter muscle and the biting force each produce a torque about the joint in a
manner depicted by the figure in below. Strategy: Find the torques produced by the two
forces by finding the portion of each force that is perpendicular to the horizontal moment
arms shown in the figure. The torque from the biting force must be the same magnitude as
the torque from the masseter muscle in order for the torques to be in equilibrium. Use the
torque produced by the biting force together with the moment arm to find the magnitude
of that force. Finally, apply Newton’s Second Law in the horizontal and vertical directions
to find the components of the force FJ that the mandible exerts on the joint? Where
FM=455N, D=0.1085m and d=0.0760
5.The triceps muscle exerts an upward force on the ulna at a point just behind the elbow
joint as indicated in the figure. Write Newton’s Second Law for torque about the elbow joint
and solve for F T where F=89N and Mg=15.6N
6.Consider the weight lifter illustrated in Fig. below, who is bent forward and lifting a
weight W0. At the position shown, the athlete’s trunk is flexed by an angle θ as
measured from the upright (vertical) position. The forces acting on the lower portion
of the athlete’s body are shown in Fig. below by considering a section passing
through the fifth lumbar vertebra. W is the total weight of the athlete, W1 is the
weight of the legs including the pelvis, (W +W0) is the total ground reaction force
applied to the athlete through the feet (at point C), FM is the magnitude of the
resultant force exerted by the erector spinae muscles supporting the trunk, and FJ is
the magnitude of the compressive force generated at the union (point O) of the
sacrum and the fifth lumbar vertebra. The center of gravity of the legs including the
pelvis is located at point B. Relative to point O, the lengths of the lever arms of the
muscle force, lower body weight, and ground reaction force are measured as a, b,
and c, respectively. Assuming that the line of pull of the resultant muscle force
exerted by the erector spinae muscles is parallel to the trunk (i.e., making an angle
θ with the vertical), determine FM and FJ in terms of b, c, θ, W0, W1, and W.
Assume that at an instant the athlete is bent so that his trunk makes an angle θ =
45_ with the vertical, and that the lengths of the lever arms are measured in terms
of the height h of the athlete and the weights are given in terms of the weight W of
the athlete as: a = 0:02h, b = 0:08h, c = 0:12h, W0 = W; and
W1 = 0:4W
7. Quadriceps tendon is inserted on the tibia 5 cm from the knee joint, and is at a 30deg
angle. Weight of the lower leg Is 48 N. Center of gravity of the lower leg is 0.20 m from the
knee joint
1. Determine Fquad required to hold the lower leg in static equilibrium
2. Determine the joint reaction force of the femur
8. Ankle Joint Forces During Walking: A person is walking with a ground reaction force of 2 times body
weight (1400 N) acting at a 20° angle to the vertical. The Achilles tendon exerts a force of 1000 N at a
60° angle to the horizontal. Calculate the joint reaction force at the ankle joint and draw free body
diagram
9.Hip Joint Forces During Walking A person is walking, and their hip joint is
flexed at a 30° angle. The weight of the lower limb is 15 kg, and its center of mass
is 30 cm from the hip joint. The hip abductor muscles exert a force at a 60° angle to
the vertical, with a moment arm of 5 cm. Calculate:
The joint reaction force at the hip joint.
The force exerted by the hip abductor muscles
Draw free body diagram
10.Spine Forces During Lifting: A person is lifting a 30 kg weight. The weight is held 50 cm from the
spine. The erector spinae muscles exert a force of 800 N at a 10° angle to the vertical. Calculate the
compressive and shear forces on the spine.
