Aims:

To describe multi-dimensional motion of rigid bodies, including rotation.

Objectives:

By the end of this course you should be able to:

• Manipulate vectors, including adding and subtracting them and converting between polar and cartesian descriptions.

• Differentiate a vector with respect to a scalar e.g. time.

• Write down Newton's Laws of Motion.

• Solve equations of motion for constant linear acceleration.

• Calculate the trajectory of projectiles under a constant gravitational force.

• Solve for the motion of bodies including frictional forces.

• Define the moment of a vector.

• Calculate the position of the centre of mass of rigid bodies of different shapes.

• Derive the equation of motion for the centre of mass of a rigid body under the action of an external force.

• Describe the optimal trajectory for a high jumper.

• Relate linear velocity and acceleration to angular velocity and acceleration for circular motion.

• Solve equations of motion for constant angular acceleration.

• Derive the kinetic energy of a rigid body rotating about a fixed axis.

• Calculate the moment of inertia of bodies of different shapes.

• State and use the perpendicular and parallel axes theorems.

• Calculate dot and cross products of given vectors.

• Calculate torques from given forces around given axes of rotation.

• Relate the angular acceleration and torque about an axis.

• Calculate the total kinetic energy of a rigid body including a rotational contribution.

• Solve equations of motion for a rigid body rotating about a moving axis using conservation of energy.

• Calculate the work done by a torque and the power generated.

• Define angular momentum and show how its component about an axis is related to the angular velocity about that axis.

• Explain what is meant by a principal axis of rotation.

• Show the relationship between torque and angular momentum.

• Describe the circumstances under which angular momentum is conserved.

• Use the concept to solve for problems in which angular velocity changes (e.g. the twizzling skater).

• Describe qualitatively the behaviour of spinning tops and gyroscopes.

• Analyse the situation when a gyroscope is undergoing steady precession.

• Solve problems similar to those done in lectures and given on the examples sheet.