By Xu B.G.

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**Extra resources for A 3-color Theorem on Plane Graphs without 5-circuits**

**Sample text**

Find the forces supporting the beam at either end. 4. 8 m from the other end. Find the tension in each wire. 5. A weightless horizontal cantilever beam projects 5 m from a vertical wall. (a) If a 15 kg mass is placed on the end of the beam furthest from the wall, find the moment of its weight about the point where the beam enters the wall. (b) If an additional two masses are placed on the beam, 10 kg at 1 m and 5 kg at 3 m from the wall, respectively, find the total moment about the point where the beam enters the wall.

This can be done by weighing the liquids in a relative density bottle, which is a small flask with a special stopper that ensures a reproducible volume each time it is filled. 2 PRESSURE Pressure is a measure of the normal (perpendicular) force acting per unit area on a surface. 90, measures 45 X 70 X 125 mm. How much does it weigh? What apparent pressure would each of its faces exert when resting on a level surface? 3 N. 9 X 103 Pa Pressure exists at any point in the body of a liquid because of the weight of liquid above it.

E. a = 0), then there can be no net vertical force acting on the object, because F = ma = O. The object exerts a downward force mg on the floor and there is an equal and opposite reaction as the floor pushes upwards on the object. Force and Motion If the lift starts to move upwards with a uniform acceleration a, then not only does it have to support the object's weight mg, it also has to support the additional force F = ma needed to make the object accelerate upwards. The floor of the lift therefore pushes upwards on the object with a total force (mg + ma) and the object experiences an apparent increase in weight.

### A 3-color Theorem on Plane Graphs without 5-circuits by Xu B.G.

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