Forces in Static EquilibriumForces in Static Equilibrium

• When an object is in static equilibrium (velocity = 0 m/s) the net force acting on the object is zero

• To solve these problems, we must break up the angled forces into their x and y components

A 20 kg sign is suspended motionless from the ceiling by two ropes, T1 and T2. Calculate the unknown tensional forces (T1 and T2).

Let’s Try an Example Together!

20 kg

T1 T2

30° 60°

Step 1: Draw a simple sketch for the forces acting on the hanging mass

T1T2

W

60°30°

Step 2: Measure the angle of each force starting from the positive x axis and rotating counterclockwise

T1T2

W

+ x axis

Step 3: Write each force with their respective angle

T1 @ 150°

T2 @ 60°

W @ 270°

T1T2

W

+ x axis

Note: Weight (Fgrav) is

always at a 270° angle

Step 4: Break each force down into its x and y components using an organized table (use cos for x and sin for y components)

ForceForce X X YY

T1 @ 150° T1cos150° T1sin150 °

T2 @ 60 ° T2cos60 ° T2sin60 °

W @ 270 ° (20)(9.8)cos270 ° (20)(9.8)sin270 °

Resultant 0 0

Step 5: Solve for any possible answers that you can in the table

Force X Y

T1 @ 150° T1cos150°

= -.866T1

T1sin150 °

= .5T1

T2 @ 60 ° T2cos60 °

= .5T2

T2sin60 °

= .866T2

W @ 270 ° (20)(9.8)cos270 °

= 0

(20)(9.8)sin270 °

= -196

Resultant 0 0

Step 6: Solve for the unknown variables (T1 and T2) by setting the sum of the x components equal to zero and/or by setting the sum of the y components equal to zero

We can do this because the system is in equilibrium (constant velocity = 0 m/s) therefore Fnet = 0

-.866T1 + .5T2 + 0 = 0

.5T2 = .866T1

.5 .5

T2 = 1.732T1

X components

Y components

Step 7: Now use the substitution method to take your answer from the x component and use it to help solve for T1 and T2

..5T1 + .866T2 + (-196) = 0

.5T1 + .866(1.732T1) + (-196) = 0

.5T1 + 1.499T1 = 196

1.99T1 = 196

T1 = 98.5 N

Step 8: Now, if necessary, solve for the other unknown force

T2 = 1.732T1

And since we know T1 = 98.5 N

T2 = 1.732(98.5)

T2 = 171 N