Week 3: Work, Energy, Momentum

Conservation of Linear Momentum

A principle stating that the total linear momentum of a closed system remains constant if no external forces act on it.

Elastic Collisions

Collisions in which both kinetic energy and momentum are conserved.

Kinetic Energy

The energy possessed by an object due to its motion, calculated as 1/2m(v²)

Momentum Conservation in 3D

In three dimensions, momentum conservation involves summing the momentum vectors across all axes (x, y, z).

Law of conservation of energy

Energy cannot be created or destroyed, only transformed from one form to another.

Mechanical energy

The sum of kinetic and potential energies in a system.

Potential Energy (U)

The stored energy of a system due to its position or configuration.

Conservative forces

Forces that do not add energy to the system and for which the work done is independent of the path taken.

Non-conservative forces

Forces that cause energy to be transferred out of the system, like friction. Depends on the path taken.

Equilibrium points

Positions where a particle experiences no net force; can be stable, unstable, or neutral.

Stable equilibrium

A position where a slight disturbance results in a restoring force back to the initial position.

Unstable equilibrium

A position where a slight disturbance causes the particle to move further away from the initial position.

Neutral equilibrium

A position where a slight disturbance does not cause any net force or movement.

Gravitational potential energy

The potential energy associated with the gravitational position of an object, given by $U = mgy$.

Potential energy diagram

A graphical representation of potential energy as a function of position, illustrating the stability of different positions.

Work-energy principle

The principle stating that the work done on a system is equal to the change in its mechanical energy.

Energy transformation

The process of changing energy from one form to another, such as from potential to kinetic energy.

Equations of motion for conservative forces

The total mechanical energy is conserved; mathematically expressed as $U+K=0$ .

Closed Path in Work

If the work done by a conservative force around any closed path is zero, it implies returning to the initial position results in no net work.

Inelastic Collision

A collision where momentum is conserved but kinetic energy is not; some kinetic energy is transformed into other forms of energy.

Perfectly Inelastic Collision

A type of inelastic collision where the two colliding objects stick together after collision and move as one mass.

Closed System

A physical system that does not exchange matter with its surroundings and is isolated from external forces.

Momentum Conservation Equation

Initial Momentum = Final Momentum

Linear Momentum

The momentum of a particle of mass moving with velocity, defined as $p=mv$ .

Impulse

The integral of force over time. Impulse is also defined as the change in momentum of an object when a force experienced by a colliding object is applied over a period of time, expressed as J.

Newton’s Second Law

The rate of change of momentum of an object is equal to the net force applied to it. This law quantifies how external forces affect the motion of an object, stating that a net force results in acceleration proportional to the force and inversely proportional to the mass of the object.

Collision

An event in which two or more objects exert strong forces on each other for a short time.

Average Force

The force experienced by an object during a collision, which can be calculated using impulse over the time of the collision.

Work-Energy Principle

The change in potential energy associated with a conservative force is equal to the negative work done by that force.

Arbitrary Zero Reference Point

A chosen point where potential energy is defined as zero, depending on the problem context.

Negative Work

Work done by transferring energy outside of a system.

Work-Kinetic Energy Theorem

States that the net work done on a particle equals the change in the particle's kinetic energy, expressed as $W=\forall K$ .

Work (W) by a variable force.

The integral of force over distance; $W=\int Fdl.$ This measures the energy transferred when a force acts on an object causing it to move. It is calculated as the product of the force applied and the distance moved in the direction of the force.

Translational Kinetic Energy

The kinetic energy associated with the translational motion of an object, dependent on mass and speed.

Change in Kinetic Energy (∆K)

The difference in kinetic energy of an object as it moves from one state to another, expressed in the work-energy theorem.

Power

The rate at which work is done by a force.

Average Power

Calculated using the formula $P_{avg} = \frac{W}{\Delta t}$.

Instantaneous Power

Defined as $P = F \cdot v$, the product of force and velocity. Also expressed as the derivative of work with respect to time.

Units of Power

Expressed in joules per second (Js$^{-1}$) or watts (W).

Work Done by Force

When a particle moves under the influence of a constant force at an angle $\phi$, the work is calculated as $W=F\cdot d=F\cos\phi\cdot d$ .

Direction of Force

In the context of power, the force is directed at some angle $\phi$ to the x-axis.

Work (W)

The transfer of energy to an object by a force acting on it.

Line Integral

A specific form of path integral used to calculate work done by variable forces. $W=\int Fdl$.

Scalar Dot Product

The product of two vectors that results in a scalar quantity. It is calculated by multiplying the magnitudes of the vectors and the cosine of the angle between them.

Unit Vectors

Vectors that have a magnitude of one and are used to determine the direction.