328 |
Week 7: Statics |
Problem 3.
lift
pivot
In the figure above, three shapes (with uniform mass distribution and thickness) are drawn sitting on a plane that can be tipped up gradually. Assuming that static friction is great enough that all of these shapes will tip over before they slide, rank them in the order they will tip over as the angle of the board they are sitting on is increased.
Problem 4.
F
M
R
h
This problem will help you learn required concepts such as:
•Static Equilibrium
•Torque (about selected pivots)
•Geometry of Right Triangles
so please review them before you begin.
A cylinder of mass M and radius R sits against a step of height h = R/2 as shown above. A
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force F is applied at right angles to the line connecting the corner of the step and the center of the cylinder. All answers should be in terms of M , R, g.
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a) Find the minimum value of |F | that will roll the cylinder over the step if the cylinder does not slide on the corner.
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b) What is the force exerted by the corner (magnitude and direction) when that force F is being exerted on the center?
Week 7: Statics |
329 |
Problem 5.
T? |
m |
M
D
θ
F? d
This problem will help you learn required concepts such as:
•Static Equilibrium
•Force and Torque
so please review them before you begin.
An exercising human person holds their arm of mass M and a barbell of mass m at rest at an angle θ with respect to the horizontal in an isometric curl as shown. Their bicep muscle that supports the suspended weight is connected at right angles to the bone a short distance d up from the elbow joint. The bone that supports the weight has length D.
a)Find the tension T in the muscle, assuming for the moment that the center of mass of the forearm is in the middle at D/2. Note that it is much larger than the weight of the arm and barbell combined, assuming a reasonable ratio of D/d ≈ 25 or thereabouts.
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b) Find the force F (magnitude and direction) exerted on the supporting bone by the elbow joint in the geometry shown. Again, note that it is much larger than “just” the weight being supported.
330 |
Week 7: Statics |
Problem 6.
W/2
D W/3
M 
D/3
W
Top view
This problem will help you learn required concepts such as:
•Force Balance
•Torque Balance
•Static Equilibrium
so please review them before you begin.
The figure below shows a mass m placed on a table consisting of three narrow cylindrical legs at the positions shown with a light (presume massless) sheet of Plexiglas placed on top. What is the vertical force exerted by the Plexiglas on each leg when the mass is in the position shown?
Week 7: Statics |
331 |
Problem 7.
M
M
T L
θ
P
This problem will help you learn required concepts such as:
•Force Balance
•Torque Balance
•Static Equilibrium
so please review them before you begin.
A small round mass M sits on the end of a rod of length L and mass m that is attached to a wall with a hinge at point P . The rod is kept from falling by a thin (massless) string attached horizontally between the midpoint of the rod and the wall. The rod makes an angle θ with the ground. Find:
a) the tension T in the string;
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b) the force F exerted by the hinge on the rod.
332 |
Week 7: Statics |
Problem 8.
Ft |
d |
H |
M |
Fb |
d |
W |
A door of mass M that has height H and width W is hung from two hinges located a distance d from the top and bottom, respectively. Assuming that the vertical weight of the door is equally distributed between the two hinges, find the total force (magnitude and direction) exerted by each hinge.
Neglect the mass of the doorknob and assume that the center of mass of the door is at W/2, H/2. The force directions drawn for you are NOT likely to be correct or even close.
Week 7: Statics |
333 |
Problem 9.
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M |
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m |
µs |
h |
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θ |
This problem will help you learn required concepts such as:
•Torque Balance
•Force Balance
•Static Equilibrium
•Static Friction
so please review them before you begin.
In the figure above, a ladder of mass m and length L is leaning against a wall at an angle θ. A person of mass M begins to climb the ladder. The ladder sits on the ground with a coe cient of static friction µs between the ground and the ladder. The wall is frictionless – it exerts only a normal force on the ladder.
If the person climbs the ladder, find the height h where the ladder slips.
334 |
Week 7: Statics |
Optional Problems
The following problems are not required or to be handed in, but are provided to give you some extra things to work on or test yourself with after mastering the required problems and concepts above and to prepare for quizzes and exams.
Optional Problem 10.
boom
T?
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m |
L |
M |
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30 |
45 |
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F?
A crane with a boom (the long support between the body and the load) of mass m and length L holds a mass M suspended as shown. Assume that the center of mass of the boom is at L/2. Note that the wire with the tension T is fixed to the top of the boom, not run over a pulley to the mass
M.
a)Find the tension in the wire.
b)Find the force exerted on the boom by the crane body.
Note:
sin(30◦) = cos(60◦) = |
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cos(30◦) = sin(60◦) = |
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sin(45◦) = cos(45◦) = |
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Week 7: Statics |
335 |
*
Optional Problem 11.
F
M
R
h
A cylinder of mass M and radius R sits against a step of height h = R/2 as shown above. A
~
force F is applied parallel to the ground as shown. All answers should be in terms of M , R, g.
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a) Find the minimum value of |F | that will roll the cylinder over the step if the cylinder does not slide on the corner.
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b) What is the force exerted by the corner (magnitude and direction) when that force F is being exerted on the center?
336 |
Week 7: Statics |