Q.Four strings pull on a single point P, which stays at rest (equilibrium). Taking the upward vertical as reference: one string pulls with a force of 2N directed up and to the left, at 45∘ from the upward vertical; a second string pulls with 1N directed up and to the right, at 45∘ from the upward vertical; a third string pulls with an unknown force F1 directed horizontally to the right (it makes 45∘ with the 1N force); and a fourth string pulls with an unknown force F2 directed vertically downward (at 90∘ from F1). Find the forces F1 and F2.
Imagine a book lying flat on a table. It doesn't move. A lamp hanging from the ceiling — still. A bridge holding up cars — motionless. What do all these have in common? They are in static equilibrium.
The word "static" means unchanging or stationary. "Equilibrium" comes from Latin aequilibrium — "equal balance." Put them together: a state where an object is completely at rest, with no tendency to start moving or rotating.
But here's the key insight: being at rest doesn't mean nothing is happening. Forces are still acting on that book — gravity pulls it down, the table pushes it up. The lamp feels tension from the ceiling pulling up and gravity pulling down. These forces are cancelling each other out perfectly.
Note
Static equilibrium is dynamic balance — forces are present, but their net effect is zero. The object "chooses" to stay still because all pushes and pulls are perfectly matched.
The Two Conditions for Static Equilibrium
For an object to be truly static (not moving or rotating), two separate things must be true simultaneously.
Condition 1: No Net Force (Translational Equilibrium)
The sum of all forces acting on the object must be zero. In vector form:
∑F=0
This means:
All upward forces equal all downward forces
All leftward forces equal all rightward forces
All forward forces equal all backward forces
If you break it into components (the standard exam approach):
∑Fx=0,∑Fy=0,∑Fz=0
Why this alone isn't enough: Imagine pushing a door at its handle — it rotates open. The forces might balance (you push, the hinges push back), but the door still moves. That's why we need the second condition.
Condition 2: No Net Torque (Rotational Equilibrium)
The sum of all torques (twisting effects) about any point must be zero:
∑τ=0
Torque depends on three things: the force applied, the distance from the pivot point, and the angle at which you push. For a force F applied at distance r from the pivot, at angle θ:
τ=rFsinθ
Watch out
A common mistake: thinking torque only matters if the object is actually rotating. Torque can be present even when nothing moves — it's just balanced by other torques. A seesaw with two kids of equal weight at equal distances is a perfect example.
Putting It All Together
For an object to be in static equilibrium:
∑F=0and∑τ=0
Both conditions must hold simultaneously. If either fails, the object will either accelerate (move in a straight line) or start rotating (or both).
A Simple Example: The Book on the Table
Consider a 2 kg book on a horizontal table. Gravity pulls down with force Fg=mg=2×9.8=19.6 N.
With P at rest, the vector sum of the four pulls is zero. Resolving into horizontal and vertical axes turns this into two simple equations, giving F1=1/2≈0.71N and F2=3/2≈2.12N.
Concept
For a point in equilibrium, ∑Fx=0 and ∑Fy=0. Take right as +x and up as +y. The 2N pull acts at 135∘ (up-left, 45∘ from vertical), the 1N pull at 45∘ (up-right), F1 along +x, and F2 along −y. Note cos45∘=sin45∘=21.
Horizontal balance (∑Fx=0)
The 2N force has a leftward component 2cos45∘; the 1N force has a rightward component 1cos45∘; F1 is fully rightward:
Concept: Resolving Concurrent Forces at Equilibrium
At equilibrium, ΣFx=0 and ΣFy=0 independently — resolve every force into horizontal and vertical components and balance each axis.
Step 1: Set up the axes and each force's components
Take right as +x, up as +y. The 2 N force (up-left, 45∘ from vertical) has components (−2cos45∘,+2sin45∘). The 1 N force (up-right, 45∘ from vertical) has components (+1cos45∘,+1sin45∘). F1 acts along +x; F2 acts along −y.
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
GUJCET 2020Set 071 markMCQ
Q.The charge density of uniformly charged infinite plane is σ. A simple pendulum is suspended vertically downward near it. Charge q0 is placed on metallic bob. If the angle made by the string is θ with vertical direction then ______.
(A) σ∝tanθ
(B) σ∝q0tanθ
(C) σ∝q0cotθ
(D) σ∝tanθq0
›Reveal solutionSolution
Balancing the horizontal electric force q0σ/2ε0 against gravity gives tanθ∝σq0, so σ∝q0tanθ.
Concept. An infinite charged plane produces a uniform field E=2ε0σ, horizontal here. The charged bob feels a horizontal force F=q0E.
Equilibrium of the pendulum. Resolving the string tension: …
Q.Two spheres having same radius and mass are suspended by two strings of equal length from the same point, in such a way that their surfaces touch each other. On depositing charge 4×10−6 C on them they repel each other in such a way that in equillibrium the angle between their strings becomes 60°. If the distance from the point of suspension to the center of the sphere is 10 cm. Find the mass of each sphere. (K=9×109 SI and g=10 ms−2.)
(A) 0.3117 kg
(B) 0.6235 kg
(C) 0.1559 kg
(D) 1.2468 kg
›Reveal solutionSolution
[!TLDR] Balancing the Coulomb repulsion against gravity gives each sphere a mass of about 0.6235 kg, option (B).
Concept
Two identical charged spheres hang so their strings make equal angles with the vertical. At equilibrium each sphere feels weight mg (down), string tension T, and horizontal Coulomb repulsion F. Resolving: Tcosθ=mg and Tsinθ=F, so tanθ=F/mg.
Solution
Charge on each sphere: q=24×10−6=2×10−6 C.
Each string makes θ=30° with the vertical; separation