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Physics 160
Biomechanics
19-1
Kinetics - Torque
τ = F·r
1. The bar (assume massless) below pivots about the far left end.
a) What is the torque exerted about this pivot, by the hanging weight below?
1.2 m
º
.8 m
2.4 kg
b) What is the torque when the pivot is moved to the far right end?
τccw = τcw
2. The vertical hand bars on a walking exerciser operate like a second class lever. The
grip is 140 cm from the pivot, while the resistance is 35 cm from the same pivot.
What effort force will pull the grip with constant velocity?
Resistance
300 N
τ = F·r sinθ
3. The distal end of a muscle has a tendinous attachment to the bone 6.2 cm from the
pivot at the joint.
a) What torque is exerted when the muscle pulls with 22 N and the tendon makes an
angle of 20° to the bone? What is it when 40° to the bone? What is it when 90° to
the bone? Is this the maximum torque?
b) When the tendon is at 35° to the bone, how hard must the muscle pull in order to
exert a torque of 1 N·m?
4. What is the maximum torque which can be produced by the biceps when it is capable
of exerting 68 N (at 2.6 cm from the joint) ?
© 2008 Camosun College Physics
Physics 160
Biomechanics
19-2
5. In question 2 above, how much torque is exerted by a person pulling horizontally with
95 N
a) When the handle is vertical? (In this position the horizontal force is
perpendicular to the force arm or handle.)
b) When the handle is 20º to one side. (In this position the horizontal force makes
an angle of 70º with the handle.)
6. Frank’s forearm is positioned horizontally so that the biceps muscle exerts a force
whose line of action is at 75º to the radius bone (also 75º to the horizontal). The
distal muscle attachment is 3.8 cm from the axis of rotation passing through the
elbow joint. The weight of the forearm alone is 8 N with the center of gravity
14 cm from the axis through the elbow.
a) What is the contractive force of the biceps in just supporting the forearm?
b) What is the total contractive force when the hand also holds a 1 kg mass
positioned 32 cm from the elbow joint?
Hint: The bicepts muscle acts as a third class lever
with a short effort arm measured from the elbow pivot
to the muscle attachment. In part a) the effort torque
balances the resistive torque due to the arm weight.
In part b) it balances two resistive torques, one for
the arm weight and one for the 1 kg mass.
Muscle force
.
Pivot
Wforearm
Wkg mass
7. Standing at the edge of a diving tower, Samantha (58 kg) raises herself up on her toes
by rotating her feet in the sagittal plane, about the ankle joint. To do this the calf
muscles pull up on the calcaneus. From the ankle pivot the horizontal distances are
3.8 cm to the calcaneus muscle attachment and 10.5 cm to the toes. Determine the
tension in the calf muscles.
8. For the paralympics Pierre’s racing wheelchair has wheels fitted with handrims
whose diameter is ½ of the diameter of the rubber wheels. How hard must
Pierre (72 kg) push forward at the top of each rim if he wants to accelerate forward
at 5.2 m/s2 ?
9. A little harder but do-able:
Basketball wheelchairs are designed with wheels having 70 cm diameter wheels and
52 cm diameter handrims. The wheels are separated by 60 cm at the floor. For a
quick rotation an athlete wants to exert a torque of 140 N·m by pushing the left and
right handrims in opposite directions. How hard must he push or pull on each rim?
© 2008 Camosun College Physics
Physics 160
Biomechanics
19-3
10. For the system below, the beam has a weight of 35 N. Calculate the total torque
about the suspension point using τ = F·r sinθ and again using τ = F·r┴ .
(Scale: 1:10)
25 N
20 N
Answers:
1. a) -18.8 N·m (cw)
b) + 9.4 N·m (ccw)
2. 75 N
3. a) 0.47 N·m, , 0.88 N·m , 1.36 N·m
b) 28.1 N
4. 1.77 N·m
5. a) 133 N·m
b) 125 N·m
6. a) 30.5 N
b) 116 N
7. 1571 N total, 785 N each
8. 374 N on each side
9. 314 N
10. 15.2 N·m (counterclockwise)
© 2008 Camosun College Physics