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Last modified: January 26, 2018
Moments and Torque
------------------
R
d3 | d2 d1
----------------------------
| ^ | |
F3 F2 F1
F3d3 = F2d2 + F1d1
R = F1 + F2 + F3
T1 T2
| d1 d2 |
----------------------------
|
F
T1d1 = T2d2
T1 + T2 = F
Ladder Problem
--------------
Consider a ladder leaning against a wall with coefficients of
friction μA and μB. Find the minimum value of θ before slippage
occurs.
Horizontal components:
FAF = FBN
μAFAN = FBN ... 1.
Vertical components:
W = FAN + FBF
W = FAN + μBFBN ... 2.
Moments around A:
(L/2)FW = LFBR
(L/2)Wcosθ = LFBNcos(90 - θ) + LFBFsin(90 - θ)
= LFBNsinθ + LFBFcosθ
check:
θ = 0 => (L/2)W = LFBF
θ = 90 => 0 = LFBN
(L/2)Wcosθ = LFBNsinθ + LμBFBNcosθ
= LFBN(sinθ + μBcosθ)
= LμAFAN(sinθ + μBcosθ)
∴ W = 2μAFAN(tanθ + μB)
From 2.
FAN + μBFBN = 2μAFAN(tanθ + μB)
From 1.
FAN + μAμBFAN = 2μAFAN(tanθ + μB)
∴ 1 + μAμB = 2μA(tanθ + μB)
∴ tanθ = (1 - μAμB)/2μA