1.3 - Forces in Equilibrium

Equilibrium and the Triangle of Forces

• An object is in equilibrium of forces when all forces acting on it give zero resultant force. Hence $F=ma=0$ and $a=0$.
• A vase at rest has weight 12 N balanced by normal reaction 12 N.
• Three forces can also balance: a suspended lamp with weight 4 N and string tensions 3 N and 5 N is one example.
• Represent three forces in equilibrium by a closed triangle of forces: side lengths represent magnitudes; successive arrow directions represent force directions.
• Equilibrium also applies to motion at uniform velocity, such as a ship pulled by two tugboats while water resistance balances the combined pull.
• A stationary box on an incline experiences weight, normal reaction and friction; a hanging picture experiences weight and two string tensions.

Balancing Forces on a Vector Force Table

• A third string force can balance two forces applied to a metal ring through pulleys.
• Equilibrium is achieved when the ring no longer touches the central rod. The balancing force’s magnitude and direction depend on the other forces and their directions.

Worked Example: Suspended Lamp

• A lamp is stationary with one tension $T=8\mathrm{N}$, directed $60^{\circ }$ above the horizontal to the left, another tension $P$ horizontally right and weight $W$ downwards.
• Horizontal and vertical forces balance separately:

• A scaled triangle provides another solution. With 1 cm representing 2 N, draw the 8 N vector 4 cm long; close the triangle with downward $W$ and rightward $P$.
• Measured sides of 2.0 cm and 3.5 cm give $P=4.0\mathrm{N}$ and $W\approx 7.0\mathrm{N}$.

Worked Example: Stationary Box on an Incline

• A box of weight 50 N rests on a plane inclined at $40^{\circ }$. Weight acts downwards, normal reaction $R$ perpendicular outwards and friction $F_R$ up the plane.
• Resolve weight into components along and perpendicular to the plane; equilibrium requires:

• With 1 cm representing 10 N, draw the weight as a 5.0 cm downward vector and close the triangle using the directions of friction and normal reaction.
• Measured friction and reaction sides of 3.2 cm and 3.8 cm give 32 N and 38 N. Calculation gives greater precision than scaled drawing.

Worked Example: Poster with Two Equal Tensions

• A 12.0 N poster hangs symmetrically from a string over a nail. Each string segment makes $35^{\circ }$ with the horizontal and has tension $T$.
• In the closed triangle, the angle opposite the weight is $70^{\circ }$ and the angle opposite each tension is $55^{\circ }$.
• Sine rule:

• Cosine rule gives the same result: