Equilibrium and work | 4A: Translational motion, forces, work, energy, and equilibrium | Chem/phys

Equilibrium

Equilibrium occurs when the vector sum of all forces acting on an object equals zero. In this state, the object remains either at rest or moves with a constant velocity, meaning there is no net acceleration. Essentially, all forces balance each other out so that the object’s motion remains unchanged.

Force causes objects to accelerate, meaning it changes their velocity or direction. Force is often represented by an arrow, where the arrow’s direction shows the direction of the force and its magnitude is often indicated beside it.
According to $F = ma$ , the unit of force is $\frac{k g \cdot m}{s ^{2}}$ .

Translational and rotational equilibrium

An object is in translational equilibrium when the net sum of all forces acting on it is zero. In this state, the object either remains at rest or moves at a constant velocity, meaning there is no acceleration present. If an object is speeding up or decelerating (which is simply acceleration in the opposite direction), it is not in equilibrium.
An object is in rotational equilibrium when the net torque acting on it equals zero. Conventionally, torques that produce counterclockwise rotation are taken as positive, while those causing clockwise rotation are negative.
In this state, the object either does not rotate at all or rotates at a constant angular velocity (or frequency). The presence of any angular acceleration—including deceleration, which is acceleration in the opposite direction—indicates that the system is not in equilibrium.

Torques, lever arms

Torque is the angular equivalent of force; it induces rotation, creating angular acceleration and changing angular velocity as well as the direction of rotation. By convention, a positive torque causes counterclockwise rotation, while a negative torque results in clockwise rotation.

A lever system consists of a rigid rod and a fulcrum, the pivot point around which rotation occurs. According to the lever arm equation, the torque remains constant along the lever arm on each side of the fulcrum. Applying a force farther from the fulcrum amplifies the effect on the load closer to the pivot; however, this advantage comes with the trade-off of requiring a longer movement of the lever.

Torque and door rotation illustrated through different force applications

Momentum

Momentum is defined as the product of an object’s mass and velocity ( $p = m \cdot v$ ) and is a vector, meaning it has both magnitude and direction. Impulse is calculated by multiplying force by the time interval during which it acts (Impulse = $F \cdot t$ ) and represents the change in momentum an object experiences.

The principle of conservation of linear momentum states that the total momentum in a closed system remains constant before and after an event. Because momentum is a vector, it is necessary to designate a positive direction when summing individual momenta. For example, the momentum of a bomb at rest is equal to the vector sum of the momenta of all its shrapnel after an explosion.

In collisions, the total momentum of objects before impact is equal to the total momentum after impact. In elastic collisions, both momentum and kinetic energy (a scalar quantity representing energy due to motion) are conserved. This means that if a ball is dropped and bounces back to its original height, or if it strikes a wall and rebounds at the same speed, the collision is perfectly elastic.

Conversely, in inelastic collisions, only momentum is conserved while some kinetic energy is transformed into other forms, such as heat or deformation. When objects stick together after colliding, the collision is classified as totally inelastic.

Work

Work is defined as the product of force ( $F$ ), distance ( $d$ ), and the cosine of the angle ( $θ$ ) between the force and the displacement, represented by the equation $W = F d cos θ$ . This equation measures the energy transferred when a force is applied over a distance. The standard unit of work is the Joule, which is equivalent to a Newton-meter ( $N \cdot m$ ) or $k g \cdot m^{2} / s^{2}$ .

The sign of work depends on the alignment of the force and the displacement. When the force and the displacement act in the same direction, the work is positive—such as when pushing a crate forward. If the force opposes the displacement, the work is negative, as seen with friction in a non-rotating system.

When the force is perpendicular to the displacement, no work is done, because the cosine of 90 is zero; for example, carrying a bucket of water horizontally involves no work on the bucket’s movement despite the effort required to hold it. The cosine function in the formula determines whether the work is positive or negative based on the angle between the applied force and the direction of movement.

The gravitational field does work in a manner that is independent of the path taken, because gravity always acts vertically. This means that any movement perpendicular to the gravitational force—such as horizontal or sideward motion—does not contribute to the work done. For instance, pushing an object up a frictionless inclined plane at a constant speed requires the same work as directly lifting it to the same height. Similarly, sliding down a frictionless inclined plane results in the same gravitational work as if the object were in free fall from that height.

Mechanical advantage

Mechanical advantage enables a small input force to produce a larger output force, allowing heavy loads to be moved more efficiently. This principle is achieved through simple machines such as lever arms and pulleys, which extend the distance over which the force is applied.

The work-kinetic energy theorem states that the work done on an object is converted into its kinetic energy; mathematically, the work ( $F \cdot d$ ) is equal to $\frac{1}{2} m v^{2}$ .
When gravity performs work on an object, the energy change is given by $m g h$ , which also equals $½ m v^{2}$ . This kinetic energy can be used to perform work, such as moving an object up an incline ( $½ m v^{2} = m g h$ ) or overcoming friction ( $½ m v^{2} = F_{f r i c t i o n} \cdot d$ ).
Power is the rate at which work is done, expressed as $P = W / t$ , and its unit is the Watt ( $W$ ), equivalent to one Joule per second.

Conservative forces

A force that does not dissipate energy as heat, sound, or light is known as a conservative force. Work done by these forces depends only on the initial and final positions, not on the path taken. Such forces are associated with potential energy; for example, a spring stores energy as spring potential energy, and gravity stores energy as gravitational potential energy.

Electromagnetic forces also fall into this category.
In contrast, non-conservative forces, like friction and the force exerted by muscles, convert mechanical energy into heat and sound that cannot be recovered.