Alternate Forms of Moment Equation (Application of Euler's 1st and 2nd Laws)
Example:A frictionless pin atAsupports a uniform bar AB of length L as shown in the figure
below.C denotes the center of mass of the
bar.The bar is released from rest
at an angle θ.
Find
the support reactions at A at this instant.
Strategy:The first
step in applying Euler’s first and/or second law is to draw a free body diagram to identify all
forces acting on the bar as shown below.Note:et x en=k
Next apply Euler’s first
lawF=m aCwhich results in
(Ft + mg sinθ)
et +(Fn ˗ mg cosθ) en=m [ actet+vC2/ρ en ]Since the bar is released
from restvC=0 which givesFt + mg sinθ=m actandFn ˗ mg cosθ=0
Next apply Euler’s 2nd law in the
formMA=IzzAα+ rAC
x m aANow|aA|=0so
MA=IzzAαfor the barIzzA=(1/3) mL2somg(L/2)sin θ k =(1/3) mL2 α k
Thusα=(3/2)(sin θ)(g/L)
.Next use kinematics to expressac in terms ofα.
aC = aA– (v2/ρ) en + α kx (– L/2)enBothaA = 0andvC = 0so
aC =(L/2) α et = (L/2)(3/2)(sin θ) (g/L)soFt + mg sinθ=m [ (3/4) g sin θ ]
Solve
forFt :Ft=(¾) mg sin θ – mg sinθ= – ¼ mg sin θandFn = mg cosθ