4.1 Forces and equilibrium

Force has magnitude and direction. Resolve it into perpendicular components, add components to find the resultant, and use Newton’s law or equilibrium equations along each axis.

Draw a force diagram, choose positive directions, and keep units consistent. Equal and opposite forces act on different bodies and should not be cancelled across a single free-body diagram.

Forces (3,4) N and (−1,2) N have resultant (2,6) N with magnitude √40 N.

A scalar magnitude alone cannot determine a resultant; direction is part of the force.

Forces combine by vector addition, so their lines of action and directions matter. A force can be resolved into components along convenient axes without changing its physical effect.

Use a consistent axis system, resolve oblique forces with sine/cosine, and interpret a negative component as opposite to the chosen positive direction.

Two 10 N forces at right angles have resultant 10√2 N, not 20 N; two opposite 10 N forces have resultant zero.

Adding magnitudes ignores angle and can overestimate the resultant.

A particle is in equilibrium when ΣF=0. Resolve horizontally and vertically (or along chosen axes), giving one scalar equation per independent direction.

Include all external forces, use geometry to express angles, and solve the component equations together. A zero horizontal resultant alone does not ensure equilibrium.

A weight supported by two symmetric strings has equal tensions; horizontal components cancel and vertical components sum to the weight.

Equilibrium does not mean no forces act; it means their vector sum is zero.

Friction acts along the contact surface and opposes relative motion or its tendency. In limiting equilibrium F=μR; otherwise F≤μR.

Draw the normal reaction and friction separately, identify the possible direction of motion, and solve force/moment equations before checking the friction limit.

A block on a rough incline has friction up the slope if it would otherwise slide down; the limiting value μR determines the largest angle for rest.

Friction is not always μR: that equality applies at limiting friction, not every static situation.

“Smooth” means the contact force is perpendicular to the surface. There is no friction component along the surface, so the unknown contact force is a single normal reaction.

Use the geometry of the surface to choose the normal direction, and do not add a friction force to a smooth-contact free-body diagram.

A particle on a smooth plane has a reaction normal to the plane and weight vertically downward; resolving these gives the motion along the plane.

Smooth does not mean weightless or force-free; it removes friction only.

A particle is in equilibrium when the resultant force is zero, so the sums of components along two independent axes both vanish.

Choose axes that simplify the geometry, include all applied forces and reactions, then solve simultaneously. A negative unknown indicates the assumed direction was opposite.

For a ring held by two strings, horizontal components cancel and vertical components add to the load; the ring remains at rest.

Equal forces are not required—different magnitudes can balance when their directions differ.

Newton’s second law is ΣF=ma for a chosen body and inertial frame. First-law equilibrium is the special case a=0; the third law pairs equal opposite forces on different bodies.

Draw one free-body diagram at a time, choose axes, and distinguish action–reaction pairs from forces that act on the same object.

A 4 kg block with resultant horizontal force 12 N has acceleration 3 m s⁻²; the reaction and weight are separate vertical forces.

The third-law partner of a normal reaction acts on the other body, so it cannot cancel the weight of the same body.