Download Pre-reading for lecture 6a: Statics

Physical Sciences 2: Lecture 6a
October 15, 2015
Pre-reading for lecture 6a:
Statics
In this lecture we study a special case of the dynamics of rigid objects covered in
previous modules. It is the important special case where the center of mass of the object
has no motion and the object is not rotating.
A surprisingly wide range of problems can be treated as if they were statics problems.
The practical importance of statics is such that all accredited engineering programs
require students to take a statics course.
Anytime you want to analyze the various forces on structures such as the cable-stayed
bridge in Boston, or the human body, statics is needed to compute the force’s magnitudes
and directions.
For a rigid object that is not moving at all we have the following two conditions:
1- The vector sum of the external forces on the rigid object must equal zero, namely
𝑭 = 0 .
When this condition is satisfied we say that the object is in translational equilibrium.
Please note that translational equilibrium doesn’t say anything about the velocity of the
object, it only says 𝑎!" = 0. In order to have static equilibrium we specify 𝑣!" = 0.
2- The sum of the external torques on the rigid object must equal zero, namely
𝝉 = 0 .
When this condition is satisfied we say that the object in in rotational equilibrium.
As in the previous case, rotational equilibrium doesn’t say anything about the angular
velocity 𝜔 of the object, it only says 𝛼 = 0. In order to have static equilibrium we
specify 𝜔 = 0.
So static equilibrium implies:
𝝉=
𝑭 = 𝑣!" = 𝜔 = 0.
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Physical Sciences 2: Lecture 6a
October 15, 2015
Whenever you solve a static equilibrium problem in this course (always in 2-D), you
should always end up with the following 3 equations:
𝜏 = 0 Torque balance about a pivot axis
𝐹! = 0 Force balance along x-axis
𝐹! = 0 Force balance along y-axis
The steps involved in solving a static equilibrium problem are:
1. Draw free-body diagram
2. Setup a coordinate system with axes
3. Apply the force balance equations
4. Apply torque balance equation (about which axis/pivot?)
5. Identify and solve for at most 3 variables
While there is only one way to write the conditions for the forces on a rigid object
summing to zero, we have a choice in the way we write the equation for the total torque.
The equation for torque does not specify the choice of the pivot axis for calculating the
torque. In general it matters a great deal which axis we pick. But when the sum of torques
about any one axis is zero and the sum of forces is zero (translational equilibrium) then
the sum of torques about any axis will give zero; so for statics problems we are free to
pick the most convenient axis for computing the net torque. Often this will be the point on
the object where several unknown forces are acting, so that the resulting set of equations
will be simpler to solve. You will practice this in lecture.
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Learning objectives: After this lecture, you will be…
1. …able to describe what we mean by static equilibrium and be able to identify if a
system is in static equilibrium.
2. …aware of physical situations where static equilibrium is relevant (e.g. structural
engineering and physiology).
3. …able to draw the free body diagram of any rigid body and identify all the forces and
their point of application, direction and magnitude.
4. …able to write all three equations for static equilibrium and identify all unknowns.
5. ...skilled in deciding which pivot axis to use for
𝜏 = 0 (this skill is non-trivial).
6. …be able to identify situations requiring more than one free body diagram.
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Physical Sciences 2: Lecture 6a
October 15, 2015
Am I getting it?
1. You hold up a spinning bicycle wheel in front of you by holding onto
the two ends of the axle. The axle is horizontal, so the wheel lies in a
vertical plane. The wheel is rotating in a direction as shown in the Fig.,
with angular velocity 𝝎 pointing to the left.
If you let go with your left hand (so that you are only holding the right
end of the axle with your right hand), the wheel:
(a) Precesses clockwise when viewed from above (i.e. center of the wheel
initially moves away from you)
(b) Precesses counterclockwise when viewed from above (i.e. center of
the wheel initially moves towards from you)
(c) This is too hard
2. What if the wheel’s angular velocity was 2𝝎? What happens to
precession then?
(a)
(b)
(c)
(d)
(e)
Precesses clockwise, but faster
Precesses counterclockwise, but faster
Precesses clockwise, but slower
Precesses counterclockwise, but slower
This is too hard…
3. What if the wheel’s angular velocity is 𝝎, but you are
on the moon where the acceleration due to gravity is
g/6. How is precession in this case?
(a)
(b)
(c)
(d)
(e)
Precesses clockwise, but faster
Precesses counterclockwise, but faster
Precesses clockwise, but slower
Precesses counterclockwise, but slower
This is too hard…
4. For those of you who completed the lab: Do you happen
to remember how fast the disc/saw was spinning?
(a) I did not do the lab yet
(c) >100 rpm
(e) >500 rpm
Point&B&
(b) >50 rpm
(d) >250 rpm
(f) I don’t remember…
Point&A&
5. The gravitational torque produced by the weight mg about pivot point A has magnitude
mgx. How about the magnitude of the net torque about pivot point B? How does it compare
to mgx?
It is…
(a) …larger
(b) …smaller
(c) …same
(e) I am a bit confused here…
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(d) …zero
Physical Sciences 2: Lecture 6a
October 15, 2015
Activity 1A: recognizing static equilibrium…
Are the objects described here in static equilibrium, dynamic equilibrium, or not in
equilibrium at all? Explain (hint: static means “nothing moves”)
(a) A 200 lb barbell is held over your head
(b) A steel beam is lifted by a crane at constant speed
(c) A steel beam is being lowered in place. It is slowing down
(d) A jet plane has reached its cruising speed and altitude
(e) A box in the back of a truck doesn’t slide as the truck stops
Activity 1B: Balancing forces and torques
The figure shows six overhead views of a uniform rod on which two or more forces act
perpendicularly to the rod? If the magnitudes of the forces are adjusted properly (but kept
nonzero), in which situations can the rod be in static equilibrium? Make sure to provide
explanations
(A)&
(B)&
(A)&
(B)&
(D)&
(E)&
(D)&
(C)&
(C)&
(E)&
Bonus: Try the same with this one
30o$
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Physical Sciences 2: Lecture 6a
October 15, 2015
Typical static problem:
Both you and your friend can hold the
same amount of weight, 35 kg, with your
forearm held out. The length of your
forearm is similar to that of your friend
(35 cm), but your biceps muscle is
attached 5 cm from the elbow joint
whereas hers is attached 8 cm from the
joint. Whose biceps exerts more force?
Solving statics problems: A strategy
!  Free-body
!  Solve
for at most 3
variables
diagram
! 
What are the forces?
! 
Which way do they point?
! 
Where do they act?
! 
Are the magnitudes
known?
force?
!  location force is acting on?
! 
! 
!  Axes
! 
!  Apply
force balance
!  Apply torque balance
!  About
!  Get
which axis?
the signs right!
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direction?
Too many variables? Not
enough equations?
Physical Sciences 2: Lecture 6a
October 15, 2015
Activity 2: kids on a seesaw
(torque must balance)
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Two kids, one heavier than the other, are playing on a seesaw. The seesaw is suspended
at point P. For this problem, assume the plank is massless.
C&
2.5 m!
a)
Draw the free-body diagram(s). (hint: for a complete picture, you need more than one FBD)
b)
Compute how far the girl has to sit in order to balance the seesaw (i.e. what is distance x)
c)
How would your answer to part (b) change if you used position A as your pivot?
Bonus: How would your answer to part (b) change if you used position C as your pivot?
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Physical Sciences 2: Lecture 6a
October 15, 2015
Activity 3: Identify the forces on Suzanne Pittman
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Your head TF is being held by her dancing partner.
Her dancing partner prevents her from falling.
a)
Draw the free-body diagram for the ballerina.
Hint: What forces act? What are their directions?
b)
Does her weight apply a torque on her about her pointy toes? If so, why isn’t she rotating
about her pointy toes? Which force balances the torque due to her weight?
c)
What is the direction of the friction force on her tippy toes. What would happen to this
Free Body if friction were not there? Would anything be different?
Bonus: If friction were not there, what would the partner have to do (in addition to
pulling her horizontally) to keep her stationary?
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Physical Sciences 2: Lecture 6a
October 15, 2015
Activity 4: Tension force in Achilles tendon
(amazing human body…)
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The Achilles tendon is attached to the heel of
the foot, as shown in the picture. We’re
interested in the situation when a person
elevates herself so that her heel is just barely off
the floor and she stands on the ball of one foot.
What can you say about the force with which
the Achilles tendon has to pull to keep your
foot in this static position?
a)
Draw the free-body diagram for the foot, assuming the Achilles tendon is purely vertical.
You may neglect the weight of the foot itself.
b)
If the person has a mass of m = 70 kg, and D = 2d (in the diagram), then what is the force
that the Achilles tendon exerts on the foot? How does it compare to your weight?
(Hint: you may also need to consider the FBD for the entire person).
Bonus: How would your answer to the previous part change if she were standing on the balls of
two feet instead of one?
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Physical Sciences 2: Lecture 6a
October 15, 2015
Activity 5: static problem (multiple FBDs)
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In this video
http://youtu.be/OirVc1SzfEk?t=41s,
2008 Olympic Gold Medalist Chen
Yibing performs the “Iron Cross.” He
holds himself up, keeping his arms
horizontal.
1. Assuming the ropes are vertical, what
is the force exerted by each rope on Chen?
2. This is the main muscle that holds him up.
When the arm is horizontal (as it is in the “Iron
Cross”) this muscle-tendon is at an angle 𝜃 from the
arm and attaches a distance 𝑅! from the shoulder
joint. Write an expression for the sum of
torques on the arm (which has length L).
Hint: (1) What is useful here? FBD of shoulder?
FBD of the arm? FBD of torso? (2) What are all the
forces on the FBD? (3) What is your pivot point to
calculate torque?
3. What is the tension in the (active) shoulder muscle? Assume distance 𝑅! is 1/20th length
of the whole arm and that 𝜃 is near 90 degrees.
Bonus! What is the tension in the (active) shoulder muscle as the distance 𝑅! go to zero?
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Physical Sciences 2: Lecture 6a
October 15, 2015
Demo:
Didn’t your mom tell you lifting with your back is bad?
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Physical Sciences 2: Lecture 6a
October 15, 2015
One-Minute Paper
Your name: _________________________________ TF: _____________________________
Names of your group members:
_________________________________
_________________________________
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Please tell us any questions that came up for you today during lecture. Write “nothing”
if no questions(s) came up for you in class from 9:30am–11am.
•
What single topic left you most confused after today’s class?
•
Any other comments or reflections on today’s class?
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