Maths in a Box (15 Parts)

Synopsis

Series of video programmes on topics in mathematics. Each programme includes and event, a question and a solution.
1: A parachutist descends. A force decelerates the body (the parachute) while a force tries to accelerate it (gravity). How to make approximations and how the resultant of the forces can be equated to the product of mass and acceleration. How to use differential equations to answer questions such as the minimum height a parachutist can jump from without injury.
2: An aircraft gathers speed and lifts off the ground. The search for approximations becomes more complex. Graphical notation emphasises the interaction and variation of the forces. Shows a variety of aeroplanes, from single-engine trainers to Concorde, and finds that a single mathematical model could fit them all.
3: An arrow is accelerated by the bowstring. Shows how to make approximations to get to a model. Looks at how geometry can help solve relatively complex problems. Is landing on an aircraft carrier the archery problem in reverse? Uses geometric similarity to show show that the solution to one problem might help to solve another.
4: A child and a man pull a sledge across the sand. The child flies a kite. Vector algebra is used to show how the resultant of a number of forces acting on a body in equilibrium and non-equlibrium can be obtained and thereby equated to the product of mass and acceleration.
5: A balloon-powered toy car rushes across the floor losing mass as it goes. Newton’s second law is considered as the rate of change of momentum rather than F=ma. Examines how impulse relates to the change in momentum, and how the rate of this change equates to the resultant force. A small experiment illustrates jet propulsion.
6: A door, a lock gate, a heart valve opens and a gymnast flic-flacs down the gym. Once the principle of a moment has been defined, the lock gates and values can be represented by the same mathematics, to a first approximation.
7: At lift-off, the rocket burns fuel as it accelerates away, and the avalanche gathers snow as it pursues the skier, while flour slides down the baking tray. In each case mass changes. The problems range from finding the finite time for a rocket to lift off the ground to calculating whether the intrepid skier is engulfed.
8: The spinning top intrigues the professional engineer. Uses animated graphics to explain the principles and show how moments relate to the rate of change of angular momentum. Graphics show the use of vector products and their reasons for ‘precession’ of the top. Explains what difference size makes to an aircraft’s aileron flaps and how much force it takes to rotate toy spaceships.
9: Bikes lean over, car wheels react going round a bend, roads are cambered. Why? Centripetal acceleration was fundamental to Newton’s ideas about why the moon orbits the earth. Explores the idea that, on the basis of velocity being a vector of quantity, circular motion must imply a force or component of a force towards the centre of the circle.
10: The ice-hockey game and skiing head-on or sideways down the hill lead to questions about the physics and the mathematical models behind sliding and toppling. Investigates the relationship between sliding and the body’s velocity as an exercise in energy conservation.
11: From bridges to artificial hip joints, ours is a three-dimensional world. Shows the use and manipulation of vectors in modelling problems.
12: Real-life examples show that vectors may take a variety of guises other than a simple force. Looks at the more complex definitions of moments using vector geometry.
13: Before pendulums, there was only the sun to help measure time. Introduces the principles of simple harmonic motion, looking at how Galileo and his contemporaries began the analysis of pendulum motion. Then shows modern-day examples.
14: Investigates how the mathematical model can be increased in complexity through the introduction of damping. Here the homogeneous linear second order differential equation appears naturally by applying simple modelling techniques. Many examples from everyday life can be modelled by this type of equation.
15: Solution techniques for general linear second order differential equations are based on the idea of adding two independent solutions together. The physics of applying a time-varying force to a body, which fundamentally exhibits simple harmonic motion, shows the way in which mathematical models provide vital information and how the independent solutions are obtained. Resonance and its effects are investigated along with examples of naturally occurring and man-made phenomena.

Language

English

Country

Great Britain

Medium

Video; Videocassette. VHS. col. 21- 50 min. each

Year of production

1999

Availability

Sale; 1999 sale: £35.00 (+VAT +p&p) each part 1999 sale: £150.00 (+VAT +p&p) set of any 5 parts

Documentation

Accompanying booklet with each part.

Uses

A-level mathematics and 1st-year undergraduate engineering.

Subjects

Mathematics; Physics

Keywords

centripetal force; differential equations; equations; force; geometry; harmonic motion; mass; mathematical modelling; momentum; pendulums; vectors; velocity

Sections

Title

Parachuting: moving bodies with constant mass

Synopsis

1: A parachutist descends. A force decelerates the body (the parachute) while a force tries to accelerate it (gravity). How to make approximations and how the resultant of the forces can be equated to the product of mass and acceleration. How to use diffe

Title

Take-off: moving bodies with constant mass

Synopsis

2: An aircraft gathers speed and lifts off the ground. The search for approximations becomes more complex. Graphical notation emphasises the interaction and variation of the forces. Shows a variety of aeroplanes, from single-engine trainers to Concorde, a

Title

Bows, arrow and aircraft carriers: moving bodies with constant mass

Synopsis

3: An arrow is accelerated by the bowstring. Shows how to make approximations to get to a model. Looks at how geometry can help solve relatively complex problems. Is landing on an aircraft carrier the archery problem in reverse? Uses geometric similarity

Title

Kites: modelling with vectors

Synopsis

4: A child and a man pull a sledge across the sand. The child flies a kite. Vector algebra is used to show how the resultant of a number of forces acting on a body in equilibrium and non-equlibrium can be obtained and thereby equated to the product of mas

Title

Impulse: moving bodies with variable mass

Synopsis

5: A balloon-powered toy car rushes across the floor losing mass as it goes. Newton's second law is considered as the rate of change of momentum rather than F=ma. Examines how impulse relates to the change in momentum, and how the rate of this change equa

Title

Doors, heart valves and flic flacs: moments

Synopsis

6: A door, a lock gate, a heart valve opens and a gymnast flic-flacs down the gym. Once the principle of a moment has been defined, the lock gates and values can be represented by the same mathematics, to a first approximation.

Title

Rockets and avalanches: moving bodies with variable mass

Synopsis

7: At lift-off, the rocket burns fuel as it accelerates away, and the avalanche gathers snow as it pursues the skier, while flour slides down the baking tray. In each case mass changes. The problems range from finding the finite time for a rocket to lift

Title

Spinning tops and aileroms: moments and angular momentum

Synopsis

8: The spinning top intrigues the professional engineer. Uses animated graphics to explain the principles and show how moments relate to the rate of change of angular momentum. Graphics show the use of vector products and their reasons for 'precession' of

Title

Bikes and cars: centripetal acceleration

Synopsis

9: Bikes lean over, car wheels react going round a bend, roads are cambered. Why? Centripetal acceleration was fundamental to Newton's ideas about why the moon orbits the earth. Explores the idea that, on the basis of velocity being a vector of quantity,

Title

Sliding and toppling: modelling forces

Synopsis

10: The ice-hockey game and skiing head-on or sideways down the hill lead to questions about the physics and the mathematical models behind sliding and toppling. Investigates the relationship between sliding and the body's velocity as an exercise in energ

Title

Modelling vectors

Synopsis

11: From bridges to artificial hip joints, ours is a three-dimensional world. Shows the use and manipulation of vectors in modelling problems.

Title

Vectors and moments

Synopsis

12: Real-life examples show that vectors may take a variety of guises other than a simple force. Looks at the more complex definitions of moments using vector geometry.

Title

Pendulum: simple harmonic motion

Synopsis

13: Before pendulums, there was only the sun to help measure time. Introduces the principles of simple harmonic motion, looking at how Galileo and his contemporaries began the analysis of pendulum motion. Then shows modern-day examples.

Title

Damping: simple harmonic motion

Synopsis

14: Investigates how the mathematical model can be increased in complexity through the introduction of damping. Here the homogeneous linear second order differential equation appears naturally by applying simple modelling techniques. Many examples from ev

Title

Resonance: simple harmonic motion

Synopsis

15: Solution techniques for general linear second order differential equations are based on the idea of adding two independent solutions together. The physics of applying a time-varying force to a body, which fundamentally exhibits simple harmonic motion,

Production Company

Name

University of Leeds Media Services

Notes

see Leeds University Television

Distributor

Name

Leeds University Television

Contact

Sally Popplewell (Sales supervisor)

Email

mediaservices@leeds.ac.uk

Web

http://mediant.leeds.ac.uk/vtcatalogue/ External site opens in new window

Phone

0113 343 2660

Fax

0113 343 2669

Address

Media Services
University of Leeds
Leeds
LS2 9JT