# PX148 Classical Mechanics & Relativity

##### Weighting: 12 CATS

By 1905, there was a successful theory (Newton's laws) describing the motion of massive bodies and there was a successful theory of light waves (Maxwell's equations of electromagnetism). But the two theories are inconsistent: in mechanics objects only move relative to each other, whereas light appears to move relative to nothing at all (the vacuum). Physicists (including Maxwell himself) had therefore assumed that there had to be some background 'ether', through which light propagated. But all attempts to detect this ether had failed. Einstein realised that there was nothing wrong with Maxwell's equations and that there was no need for an ether. Newtonian mechanics itself was the problem. He proposed that the laws of classical mechanics had to be consistent with just two postulates, namely that the speed of light is a constant and that all frames of reference are equivalent. These postulates forced Einstein to reject previous ideas of space and time and led directly to the special theory of relativity.

This module studies Newtonian mechanics emphasizing the conservation laws inherent in the theory. These have a wider domain of applicability than classical mechanics (for example they also apply in quantum mechanics). It also looks at the classical mechanics of oscillations and of rotating bodies. It then explains why the failure to find the ether was such an important experimental result and how Einstein constructed his theory of special relativity. The module covers some of the consequences of the theory for classical mechanics and some of the predictions it makes, including: the relation between mass and energy, length-contraction, time-dilation and the twin paradox.

Aims:
To revise A-level classical mechanics and to develop the theory using vector notation and calculus. To introduce special relativity. To cover material required for future physics modules.

Objectives:
At the end of the module, you should be

• Able to solve F =d p /dt for a variety of simple cases;
• Familiar with the concepts of potential and kinetic energy;
• Able to recognise and solve the equations of forced and damped harmonic motion;
• Able to solve problems involving torque and angular momentum;
• Able to explain the transformation between inertial frames of reference (Lorentz transformation) and to work through illustrative problems.

Syllabus:
Forces, interactions and Newton's Laws of Motion

Applying Newton's Laws - equilibrium, dynamics of particles, friction and dynamics of circular motion Work and kinetic energy.

Potential energy and energy conservation.

Conservation of momentum, elastic collisions, centre of mass

Rotation of rigid bodies - angular velocity and acceleration, Dynamics of rotational motion, conservation of angular momentum

Hooke's law, equation of motion for a mass attached to a spring on a frictionless plane. Solutions for shm. Energy in shm. The pendulum, departures from shm for large amplitude. Complex notation. Damping: critical and under-/over-damping. Forced oscillations.

Motion as seen by different observers. Galilean Transformation of Velocities. Inertial frames of reference

The Michelson Morley experiment. The universality of the speed of light. The meaning of simultaneity.

Einstein's postulates: Lorentz transformation, Inverse Lorentz transformation and invariants. Length Contraction and Time Dilation, Doppler Effect.

Einstein's energy and mass relation, energy and momentum of elementary particles.

Minkowski diagrams - graphical representation of past/present/future.

Commitment: 30 Lectures + 10 problems classes

Assessment: 2 hour examination