Equilibrium occurs when the vector sum of all forces acting on an object equals zero. In this state, the object either stays at rest or moves with a constant velocity, so there is no net acceleration. In other words, the forces balance, and the object’s motion doesn’t change.

Force causes objects to accelerate, meaning it changes their velocity or direction. Force is often represented by an arrow: the arrow points in the direction of the force, and its magnitude is shown by the arrow’s length or by a number written next to it.
According to $F = ma$ , the unit of force is the Newton (N), which 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. Then the object either remains at rest or moves at a constant velocity, so there is no acceleration. If an object is speeding up or decelerating (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. By convention, torques that produce counterclockwise rotation are taken as positive, while those causing clockwise rotation are negative.

In this state, the object either doesn’t rotate or rotates at a constant angular velocity (or frequency). Any angular acceleration - including deceleration - means the system is not in equilibrium.

Torques, lever arms

Torque is the angular equivalent of force. A torque tends to make an object rotate, producing angular acceleration and changing angular velocity (and potentially the direction of rotation). By convention, a positive torque causes counterclockwise rotation, while a negative torque results in clockwise rotation.

The unit of torque is the newton-metre (N⋅m), or $k g m^{2} se c^{-} 2$ .

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 increases the torque on the load closer to the pivot. The trade-off is that the point where you apply the force must move a longer distance.

A diagram illustrating torque and door rotation through different force applications

Torque and door rotation illustrated through different force applications

Momentum

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

The unit of momentum is the newton-second, or N*m/s, or $k g \cdot m / s$ .

The principle of conservation of linear momentum states that the total momentum in a closed system stays constant before and after an event. Because momentum is a vector, you must choose a positive direction before adding 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 the product of force ( $F$ ), distance ( $d$ ), and the cosine of the angle ( $θ$ ) between the force and the displacement: $W = F d cos θ$ . This equation measures the energy transferred when a force acts 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 displacement point in the same direction, the work is positive (for example, pushing a crate forward).
If the force opposes the displacement, the work is negative (for example, 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 does no work on the bucket’s horizontal motion, even though you still have to exert force to hold it up. The cosine term in the formula determines whether the work is positive, negative, or zero based on the angle between the force and the motion.

The gravitational field does work in a way that is independent of the path taken because gravity always acts vertically. Motion perpendicular to gravity (horizontal or sideways motion) doesn’t contribute to gravitational work. For instance, pushing an object up a frictionless inclined plane at a constant speed requires the same work as lifting it straight up to the same height. Similarly, sliding down a frictionless inclined plane involves the same gravitational work as free fall from that height.

Mechanical advantage

Mechanical advantage lets a small input force produce a larger output force, making it easier to move heavy loads. Simple machines such as lever arms and pulleys do this by increasing the distance over which the input force is applied.

The work-kinetic energy theorem states that the work done on an object becomes 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 called a conservative force. Work done by a conservative force depends only on the initial and final positions, not on the path taken. Conservative 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.

Equilibrium

Occurs when vector sum of all forces equals zero
No net acceleration; object at rest or constant velocity
Force unit: Newton (N), $F = ma$

Translational and rotational equilibrium

Translational: net force = 0; no linear acceleration
Rotational: net torque = 0; no angular acceleration
- Counterclockwise torque = positive; clockwise = negative

Torques, lever arms

Torque: angular equivalent of force; unit = N·m
Lever: rigid rod + fulcrum; torque increases with distance from fulcrum
Lever arm equation: torque is constant along lever arm

Momentum

Momentum ( $p$ ): mass × velocity; vector quantity
Impulse: force × time; equals change in momentum
Conservation: total momentum in closed system remains constant
- Elastic collisions: momentum and kinetic energy conserved
- Inelastic collisions: only momentum conserved; kinetic energy not conserved

Work

$W = F d cos θ$ ; unit = Joule (J)
Positive work: force and displacement same direction
Negative work: force opposes displacement
No work: force perpendicular to displacement ( $cos 9 0^{\circ} = 0$ )
Gravitational work: path-independent; depends only on vertical displacement

Mechanical advantage

Simple machines (levers, pulleys): increase output force by increasing input distance
Work-kinetic energy theorem: $W = \frac{1}{2} m v^{2}$
Gravitational work: $m g h = \frac{1}{2} m v^{2}$
Power: $P = W / t$ ; unit = Watt (W)

Conservative forces

Path-independent; associated with potential energy
Examples: gravity, springs, electromagnetic forces
Non-conservative forces (e.g., friction): dissipate energy as heat, sound

Equilibrium and work

Equilibrium

Translational and rotational equilibrium

Torques, lever arms

The unit of torque is the newton-metre (N⋅m), or $k g m^{2} se c^{-} 2$ .

A diagram illustrating torque and door rotation through different force applications

Momentum

The unit of momentum is the newton-second, or N*m/s, or $k g \cdot m / s$ .

Work

The sign of work depends on the alignment of the force and the displacement:

When the force and displacement point in the same direction, the work is positive (for example, pushing a crate forward).
If the force opposes the displacement, the work is negative (for example, friction in a non-rotating system).

Mechanical advantage

The work-kinetic energy theorem states that the work done on an object becomes 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

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.

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4. Chem/phys

4.1. Translational motion, forces, work, energy, and equilibrium

Equilibrium and work

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Equilibrium

Translational and rotational equilibrium

Torques, lever arms

The unit of torque is the newton-metre (N⋅m), or $k g m^{2} se c^{-} 2$ .

A diagram illustrating torque and door rotation through different force applications

Momentum

The unit of momentum is the newton-second, or N*m/s, or $k g \cdot m / s$ .

Work

The sign of work depends on the alignment of the force and the displacement:

When the force and displacement point in the same direction, the work is positive (for example, pushing a crate forward).
If the force opposes the displacement, the work is negative (for example, friction in a non-rotating system).

Mechanical advantage

The work-kinetic energy theorem states that the work done on an object becomes 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

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.

Equilibrium

Occurs when vector sum of all forces equals zero
No net acceleration; object at rest or constant velocity
Force unit: Newton (N), $F = ma$

Translational and rotational equilibrium

Translational: net force = 0; no linear acceleration
Rotational: net torque = 0; no angular acceleration
- Counterclockwise torque = positive; clockwise = negative

Torques, lever arms

Torque: angular equivalent of force; unit = N·m
Lever: rigid rod + fulcrum; torque increases with distance from fulcrum
Lever arm equation: torque is constant along lever arm

Momentum

Momentum ( $p$ ): mass × velocity; vector quantity
Impulse: force × time; equals change in momentum
Conservation: total momentum in closed system remains constant
- Elastic collisions: momentum and kinetic energy conserved
- Inelastic collisions: only momentum conserved; kinetic energy not conserved

Work

$W = F d cos θ$ ; unit = Joule (J)
Positive work: force and displacement same direction
Negative work: force opposes displacement
No work: force perpendicular to displacement ( $cos 9 0^{\circ} = 0$ )
Gravitational work: path-independent; depends only on vertical displacement

Mechanical advantage

Simple machines (levers, pulleys): increase output force by increasing input distance
Work-kinetic energy theorem: $W = \frac{1}{2} m v^{2}$
Gravitational work: $m g h = \frac{1}{2} m v^{2}$
Power: $P = W / t$ ; unit = Watt (W)

Conservative forces

Path-independent; associated with potential energy
Examples: gravity, springs, electromagnetic forces
Non-conservative forces (e.g., friction): dissipate energy as heat, sound

Equilibrium and work

Equilibrium

Translational and rotational equilibrium

Torques, lever arms

The unit of torque is the newton-metre (N⋅m), or $k g m^{2} se c^{-} 2$ .

A diagram illustrating torque and door rotation through different force applications

Momentum

The unit of momentum is the newton-second, or N*m/s, or $k g \cdot m / s$ .

Work

The sign of work depends on the alignment of the force and the displacement:

When the force and displacement point in the same direction, the work is positive (for example, pushing a crate forward).
If the force opposes the displacement, the work is negative (for example, friction in a non-rotating system).

Mechanical advantage

The work-kinetic energy theorem states that the work done on an object becomes 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

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.