Q.State the conditions for the equilibrium of a particle under concurrent forces.

For rotational equilibrium, the torques (moment = force x perpendicular distance from the pivot) on both sides of the seesaw must be equal; a bigger weight needs a smaller distance to produce the same torque as a smaller weight at a larger distance.

For a seesaw balanced about its pivot (fulcrum), the principle of moments requires:

Weight1 x distance1 = Weight2 x distance2

If person 1 (heavier) has a larger weight W1, and person 2 (lighter) has weight W2 < W1, sitting at distance d2 from the pivot, then to balance:

d1 = (W2 x d2) / W1

…

Tension is smallest when the strings are exactly vertical, because the whole tension then acts along the direction opposing gravity with no component 'wasted' sideways.

By symmetry, each string carries the same tension $T$ and its two horizontal components cancel, so only the vertical components support the weight $W$ of the frame:

$2T\cos\theta = W \quad\Rightarrow\quad T=\frac{W}{2\cos\theta}$

where $\theta$ is the angle each string makes with the vertical.

• In (a) the strings splay outward (diverge), so $\theta>0$, hence $\cos\theta<1$ and $T>W/2$.
• In (b) the strings are vertical, so $\theta=0$, $\cos\theta=1$, giving the minimum possible tension $T=W/2$. …

A floating object at rest is in equilibrium — the upward buoyant force exactly balances its downward weight, giving zero net force.

By Newton's first law, an object at rest stays at rest only if the net force on it is zero. A cork floating (not sinking or rising) on water has:

• Weight (mg) acting downward
• Buoyant force (upthrust from displaced water) acting upward, exactly equal to mg …

A body moves with uniform velocity only when the net force on it is zero, so the third force must exactly cancel the sum of the first two.

Given: F1 = 2i - 5j, F2 = 3i - 4j.

For the velocity to become (and stay) uniform, the net force on the body must be zero (Newton's first law - no acceleration means no net force): …

A floating object is in translational equilibrium: the upward buoyant force (from displaced water) exactly balances the downward weight, so the net force is zero -- consistent with it floating at a fixed depth rather than accelerating up or down.

A cork floating (at rest, at a constant level) on the surface of water is in mechanical equilibrium. Two forces act on it:

1. Its weight, mg, acting downward.
2. The buoyant (upthrust) force from the displaced water, acting upward, equal in magnitude to the weight of water displaced (Archimedes' principle).

Since the cork is floating steadily (not sinking or rising), these two forces must be equal and opposite, so the net force on it is: …

A particle is in equilibrium only when the vector sum of every external force on it is zero.

A particle is said to be in (translational) equilibrium when it has no acceleration — it is either at rest or moving with constant velocity. From Newton's second law, $F_{net} = ma$. If $a = 0$, then $F_{net}$ must be $0$. A 'constant' nonzero force, a 'positive' force, or a 'negative' force would each still produce a nonzero …