Wize University Statics Textbook (Master) > Equilibrium of a Particle

 Definitions
Particle: An object with negligible size (not necessarily negligible mass) – so all forces act at a single point
Body: An object with non-negligible size – meaning we have to consider the effect of moments
Equilibrium: When an object has no tendency to translate or rotate
Loading on a body: All forces and couple moments acting on a body including distributed loads, but excluding reaction forces.

In the first section of the course, we will be mainly focusing on particle equilibrium, or the equilibrium of a single point in space. The general equation for equilibrium is:

ΣF⃗ =0\Sigma \vec{F}^{\,} =0ΣF=0
This is a vector equation

ΣFx=0\Sigma F_x=0ΣFx​=0
ΣFy=0\Sigma F_y=0ΣFy​=0
ΣFz=0\Sigma F_z=0ΣFz​=0
If a particle is not in equilibrium, then the resultant force can be represented as a summation of all forces acting on the particle.

FR⃗ =ΣF⃗ =ΣFxi+ΣFyj+ΣFzk\vec{F_R}^{\,}=\Sigma \vec{F}^{\,}=\Sigma F_xi+\Sigma F_yj+\Sigma F_zkFR​​=ΣF=ΣFx​i+ΣFy​j+ΣFz​k

Before dealing with forces, we will first consider the Free Body Diagram (FBD).

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If the tension in BC is 500N, determine the tension in BD and AB, and determine
the stretch in the spring.

Solution:


ΣFy=0\Sigma F_y=0ΣFy​=0
500 sin⁡45−FBDsin⁡36.9=0500\ \sin45-F_{BD}\sin36.9=0500 sin45−FBD​sin36.9=0
FBD=589NF_{BD}=589NFBD​=589N

Fs=kΔlF_s=k\Delta lFs​=kΔl                                            ΣFx=0\Sigma{F}_x=0ΣFx​=0
Δl=Fsk=825N800 N/m\Delta l=\frac{F_s}{k}=\frac{825N}{800\ N/m}Δl=kFs​​=800 N/m825N​                           500cos45+589cos36.9=Fs500cos45+589cos36.9=F_s500cos45+589cos36.9=Fs​     
Δl=1.03m\Delta l=1.03mΔl=1.03m                                        Fs=825NF_s=825NFs​=825N

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Determine the mass of A needed to hold the system at equilibrium.

Solution:
TBC=40g=392.4 NT_{BC}=40g=392.4\ NTBC​=40g=392.4 N

ΣFy=0\Sigma{F}_y=0ΣFy​=0
392.4sin30−TAE=0392.4sin30-T_{AE}=0392.4sin30−TAE​=0
TAE=196.2NT_{AE}=196.2NTAE​=196.2N
TAE=MAgT_{AE}=M_AgTAE​=MA​g
196.2N=MA(9.81m/S2)196.2N=M_A(9.81m/S2)196.2N=MA​(9.81m/S2)
MA=20kgM_A=20kgMA​=20kg

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