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: Pivot the bar about its center stretching bands of equal length are unstretched the bands equal amounts as shown
Gr in the middle not possible to pivot with Where would you apply a single force to Pi pivot the bar to this position? 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: Pivot the bar about its center stretching bands of equal length are unstretched the bands equal amounts as shown
1 Gr What is the smallest number of 2 Pi forces which can be used to maintain this pivoted position? 3 Bl
Can you produce this motion with more than one combination of Yes Gr forces? No Pi Couples in 2-D
Unloaded Configuration: Pivot the bar about its center bands of equal length stretching the bands equal are nnstretched amounts as shown
Must the forces be acting Yes Gr parallel to the rubber bands? No Pi Couples in 2-D
Unloaded Configuration: Pivot the bar about its center stretching bands of equal length are unstretched 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 d F F
F F
d d F F
F F
All pairs of forces exerted by fingers produce statically equivalent effect Couples in 2-D
Unloaded Configuration: Pivot the bar about its center bands of equal length stretching the bands equal are unstretched 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 d R F F
a M F F R
ΣM|c = Fd - Ra = 0 ΣM|c = Fd - M = 0 Ra = Fd 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 Couples in 2-D F 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