Couples in 2-D
Approximately 45°
Initial configuration:
• bar on a horizontal plane
• equal sized bands
• straightened out
• just about to be stretched
Couples in 2-D
45°
Unloaded Configuration:
bands of equal length are unstretched
Where would you apply a single force to
pivot the bar to this position?
Pivot the bar about its center stretching
the bands equal amounts as shown
Gr
in the middle
Pi
not possible to pivot with
single force
Bl
at the end
Ye
at the peg
Couples in 2-D
F
F
Bar is to be pivoted about center (bands stretch equal amounts)
It is not possible to balance the pair of equal and opposite
forces F (exerted by the bands) with a single force !
Pi
Couples in 2-D
Unloaded Configuration:
bands of equal length are unstretched
Pivot the bar about its center stretching
the bands equal amounts as shown
What is the smallest number of
forces which can be used to maintain
this pivoted position?
Can you produce this motion with
more than one combination of
forces?
1
Gr
2
Pi
3
Bl
Yes
No
Gr
Pi
Couples in 2-D
Unloaded Configuration:
bands of equal length
are nnstretched
Must the forces be acting
parallel to the rubber bands?
Pivot the bar about its center
stretching the bands equal
amounts as shown
Yes
No
Gr
Pi
Couples in 2-D
Unloaded Configuration:
bands of equal length are unstretched
Pivot the bar about its center stretching
the bands equal amounts as shown
What can you do to decrease the magnitude of a pair of
forces needed to produce this motion?
Couples in 2-D
Given the band forces F, what else must be applied
to the bar to maintain it in this position?
d
F
F
ΣFy = F - F = 0:
no net force is needed
ΣM|c = F(d/2) +F(d/2) = Fd ≠ 0:
must apply a moment (-Fd)
Couples in 2-D
d
F
F
F
F
d
F
d
d
F
F
F
All pairs of forces exerted by fingers produce statically equivalent effect
Couples in 2-D
Unloaded Configuration:
bands of equal length
are unstretched
Pivot the bar about its center
stretching the bands equal
amounts as shown
Load the body so it pivots using only the nutdriver.
How are forces are actually being applied to the nut, and
how are they different and similar to a pair of forces
which produces the same motion?
Couples in 2-D
d
R
d
F
F
M
a
F
R
ΣM|c = Fd - Ra = 0
Ra = Fd
F
ΣM|c = Fd - M = 0
M = Fd
Couples in 2-D
d
F
M
F
• even a couple M corresponds to two or more forces
(on the corners of the hexagon here)
• we are just not interested in sorting those detailed forces out
• their net effect – the moment they produce - is all that matters
Moment of the couple is same regardless of the point used to
calculate the sum of moments
F
Couples in 2-D
F
F
F
F
F
F
F
F
M
F
All pairs of forces exerted by fingers produce statically equivalent effect to the
couple exerted by the nutdriver