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Extra resources for Applied Mathematics by Example: Exercises

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The string breaks when the tension in it exceeds 12 mg. Show that at this instant 5 θ = arcsin(3/5). Show also that M hits the trunk of tree OG after a further time 3 8 5L . 2g Find the distance below O that the impact occurs. 7. 6 m/s2 . It is set running with the hands indicating 12 o’clock. What time does it indicate one (true) hour later? 8. An executive in a tower block stares out of a window 2 m wide. Immediately outside the centre of the window there appears a window cleaner suspended by long ropes from the top of the building.

What value of m is required to achieve this? 3. Mr D is designing a centreboard for his dinghy, which is to be cut out of a uniform sheet of metal. The proposed shape as shown is enclosed between the lines x = 0, x = 1, x = y, y = 3. x=1 y=3 y=x y=1 What is the total area of the centreboard? Where is the centre of gravity of the rectangular part of the centreboard (between x = 0, x = 1, y = 1, y = 3)? Where is the centre of gravity of the triangular part (enclosed between x = 0, x = y, y = 1)? Where is the centre of gravity of the complete centreboard?

Each step has a vertical rise of 17 cm. What is the power generated by Mr C? 11. An escalator 40 metres long inclined at an angle of 30◦ to the horizontal has a capacity of 100 passengers per minute. Assuming the average passenger has a mass of 70 kg, what is the power required to drive the escalator? 12. A car of mass 800 kg is travelling on a level road at a speed of 20 m/s against a resistance force of 250 newtons. 5 m/s2 . Assuming no resistance forces, calculate (a) the tractive force of the engine at this instant and (b) the power output of the engine.

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