Mechanics Course - 2024-25 Autumn Term

The study of how objects move in response to forces.

A set of rules that describe how the universe works, often expressed mathematically.

The branch of mathematics that deals with rates of change.

A quantity that stays the same in a closed system, even when other things change.

The idea that the laws of physics are the same for everyone, regardless of their motion.

The idea that objects at rest tend to stay at rest, and objects in motion tend to stay in motion.

The study of the very small, where the laws of classical mechanics no longer apply.

The study of very large objects, like galaxies and the universe.

The time it takes for a moving object's velocity to decrease to approximately 37% of its initial value.

A quantity that represents the product of an object's initial velocity (v0) and the time constant ( τ) of its motion.

The work–energy theorem states that the work done on an object by a force is equal to the change in its kinetic energy.

The velocity of an object is often a function of time, (v(t)). However, in specific situations, the velocity can be expressed as a function of its position (v(x)).

A force can act on an object and can depend on the object's position, velocity, and time.

This rule allows us to calculate the acceleration of an object by relating its velocity and position functions.

The mass used in equations describing the internal motion of a system. It simplifies calculations by treating the system as if it were a single object with this 'reduced' mass.

The motion of a system's center of mass is determined by external forces, and the total mass of the system is used.

A brief interaction between two or more bodies resulting in a significant change in their motion.

An internal force that acts between the parts of a system, causing changes within the system.

An external force that acts on the entire system, causing the system to move as a whole.

In classical mechanics, this principle states that a particle's kinetic energy is always greater than or equal to zero. This implies that the particle can never have negative kinetic energy.

The regions where the total energy (E) of a particle is greater than or equal to the potential energy (U(x)) are considered viable regions for the particle to move. It represents the areas where the particle has sufficient energy to exist.

When the total energy (E) of a particle is less than the potential energy (U1) at a specific point, the particle is confined to a single region and cannot cross the potential barrier. It will oscillate back and forth within this region.

When the total energy (E) lies between two potential energy levels (U1 and U2), the particle can exist in two separate regions. It can move freely within each region but cannot cross the potential barrier between them.

When the total energy (E) is greater than the highest potential energy level (U2), the particle has enough energy to move freely across all potential barriers and exists in a single, continuous viable region.

The behavior of a particle in a potential well, oscillating between two points (A and B) determined by its total energy (E) and potential energy (U(x)) is similar to a ball rolling back and forth in a valley.

When a particle oscillates within a potential well, and the points of turning (A and B) are close enough to the minimum of the potential function (x0), the potential function can be approximated as a quadratic function. This simplified representation simplifies the analysis of the particle's motion.

The analogy of a ball rolling on a hilly landscape with height h(x) = U(x)/mg can be used to understand the motion of a particle in a potential U(x). The gravitational potential energy of the ball represents the potential energy of the particle.

The change in potential energy near a minimum is approximately quadratic, meaning it resembles the shape of a parabola. This means that the force acting on the object can be approximated as a linear restoring force, similar to a spring.

A Taylor series expansion is a way to represent a function as an infinite sum of terms, each of which is a power of the variable. By expanding the potential function around its minimum, we can get a good approximation of its behavior near that point.

The first derivative of the potential function, evaluated at the minimum, is zero because the slope of the potential function is zero at its minimum. This means that the force is also zero at the minimum point.

Higher order terms in the Taylor series expansion become smaller as the distance from the minimum point decreases. This is why the quadratic term dominates for small oscillations.

A V-shaped notch is an example of a potential function that does not become quadratic when you zoom in on its minimum. This is because the curvature of the potential function at the minimum is infinite.

If a function has a Taylor series expansion about its minimum and a non-zero value of the second derivative at the minimum, then it is approximately quadratic on a small enough scale.

Hooke's law states that the force exerted by a spring is proportional to the displacement from its equilibrium position. This law applies to small oscillations about the minimum of a potential well.

Small oscillations about the equilibrium position of a potential well are often simple harmonic, meaning that the force is proportional to the displacement from equilibrium and the motion is sinusoidal.

The point that represents the average position of all the mass in a system, weighted by their respective masses.

The equation that states the rate of change of the centre of mass momentum is equal to the net external force applied to the system.

The equation that describes the motion of one object relative to another in a two-body system, where the system's total mass is reduced to a single effective mass.

In a two-body system, the reduced mass reflects the effective mass that participates in the internal motion between the two bodies.

The motion of the individual particles within a system, independent of the overall motion of the system.

The total force acting on a system, calculated as the vector sum of all the external forces acting on each individual particle within the system.