Physics 106 Week 7:
Equilibrium I - Basics
SJ 7th Ed.: Chap 12.1 to 3
• The equilibrium conditions
• Types of mechanical equilibrium
• Center of gravity
–
–
–
–
Definition
Methods for finding CG
CG versus mass center
Examples
• Solving equilibrium problems
– Some useful theorems
– Problem solving rules and methods
– Sample problems
Today
Rules for solving equilibrium problems
G
First Condition:
Fnet = 0
G
τnet = 0
Second Condition:
Choose any
y convenient axis for torque
q calculation.
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Problem PP10603-11: A meter stick balances
horizontally on a knife-edge at the 50.0 cm mark.
With two 5.0 g coins stacked over the 12.0 cm
mark, the stick is found to balance at the 45.5 cm
mark. What is the mass of the meter stick?
Center of gravity for a sledge hammer
‰ 7.1. The center of gravity for a sledge hammer lies on the centerline of
the handle, close to the head, at the X mark. Suppose you saw across
the handle through the center of gravity, cutting the ax in two pieces.
You then weigh both pieces. Which of the following will you find?
x
A)
B)
C)
D)
The handle piece is heavier than the head piece
The head piece is heavier than the handle piece
The two pieces are equally heavy
The comparative weights depend on more information
2
iClicker quiz
A
All forces have the same
magnitude.
Which force has the greatest
magnitude
i d off torque with
ih
respect to the pivot?
Pivot
B
(a) A
(b) B
C
(c) C
(d) All forces have the same torque
(e) Not enough information
Example: Weight distribution between front and rear wheels of a car
PP10603-08: A car whose mass = 1360 kg has 3.05 m between its front
and rear axles. Its center of gravity is located 1.78 m behind the front
axle. The car is stationary on level ground. Find the magnitude of
the force from the ground on each front and rear wheel (assuming
equal forces on both sides of the car).
L= 3.05 m
2FR
2FF
mg
d = 1.78 m
3
Physics 106 Week 8
Equilibrium II
SJ 7th Ed.: Chap 12.1 to 3
• More Static equilibrium example problems
Find the forces on the pivot
‰ 8.1. The sketches show four overhead views of uniform disks that can
slide or rotate on a frictionless floor. Three forces act on each disk,
either at the rim or at the center. Which disks are in equilibrium?
2
1
3
F
F
4
F
2F
F
2F
F
3F
2F
F
A)
B)
C)
D)
E)
2F
F
1, 2, 3, 4
1, 3, 4
3
1, 4
3, 4
G
Fnet = 0
G
τnet = 0
4
Example 1: Massless beam supporting a weight
The 2.4 m. long weightless beam shown
in the figure is supported on the right
by a cable that makes an angle of 50o
with the horizontal beam. A 32 kg mass
hangs from the beam 1.5 m from the
pivot point on the left.
x
m = 32 kg
θ = 50 o , x = 1.5 m,
a) Calculate the torque caused by the
hanging mass.
b) Determine the cable tension needed
L = 2.4 m to produce equilibrium
c) Find the x and y components of the
force at the pivot point A.
A
Example 2: Beam with a mass, supporting a weight
The 2.4 m long 20 kg uniform beam
shown in the figure is supported on the
right by a cable that makes an angle of
50o with the horizontal beam. A 32 kg
mass hangs from the beam 1.5 m from
the pivot point on the left.
x
Determine the cable tension needed to
produce equilibrium
m = 32 kg
θ = 50 o , x = 1.5 m,
L = 2.4 m
5
Example 3:
A uniform beam, of length L and mass m = 1.8 kg, is at rest with its ends
on two scales (see figure). A uniform block, with mass M = 2.7 kg, is at
rest on the beam, with its center a distance L / 4 from the beam's left end.
What do the scales read?
Example 4:
A safe whose mass is M = 430 kg is
hanging by a rope from a boom with
dimensions a = 1.9 m and b = 2.5 m.
The boom consists of a hinged
beam and a horizontal cable that
connects the beam to a wall. The
uniform beam has a mass m of 85
Fv
kg; the mass of the cable and rope
are negligible.
Tc
Tr
mg
Fh
(a) What is the tension Tc in the
cable; i. e., what is the magnitude of
the force Tc on the beam from the
horizontal cable?
(b) What is the magnitude of the
force at the hinge?
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