Specimen M1 Paper

1. A tree branch is inclined at 22 degrees to the horizontal. A merry man resting on it is just on the point of slipping. Modelling the outlaw as a particle and the tree branch as a rough plane, find
   1. the coefficient of friction between the branch and the merry man;
   2. the angle the branch would make to the horizontal if the coefficient of friction were twice as large and the merry man were again on the point of slipping;
   3. the coefficient of friction between the branch in part (b) and a horn of mead resting on it just on the point of slipping (a horn of mead rests on a surface at three points), were the branch to break from the tree without rotating at all and fall at an acceleration of 5 m/s/s.
2. An evil henchman winds a crossbow up against a constant force of 500 N. If the crossbow's cord is wound up 0.5 m,
   1. what is the maximum height above the crossbow a bolt of 0.2 kg would reach if fired vertically upwards?
3. A drawbridge of length 5 m is pivoted at one end. 5 m directly above the pivot a rope runs over a pulley. The free end of the rope is attached to the other end of the drawbridge. The mass of the drawbridge is 500 kg. Find
   1. the tension required in the rope to balance drawbridge at the horizontal;
   2. and when the drawbridge is elevated slightly, is more or less tension required to balance it?
4. A reeve of the Sheriff of Nottingham's runs down a very shallowly inclined hill in pursuit of the peasant who has stolen his horse. The mass of the reeve with various military accessories is 120 kg. The resistance to his motion less the force from his running propelling him down the hill is 0 N. After 30 seconds the reeve reaches the bottom of the hill, where he must clear a stream. He jumps upwards, immediately changing the vertical component of his velocity to 1 m/s, and just clears the stream which is 3 m across. Modelling the hill as a smooth inclined plane, the reeve as a particle, and assuming the reeve started from rest at the top of the hill, find
   1. the height of the hill;
   2. and its angle of inclination.
5. The Sheriff's chief executioner is ordered to hang Friar Tuck (who weighs in at a round 100 kg). The force required to break the good friar's neck is 1000 N. If more than this force is exerted, the friar will be decapitated. If less is exerted the friar will not be killed immediately. When the friar comes to the end of the drop he will be brought to rest in 0.2s.
   1. How long should the drop be?
   Name two modelling assumptions you have made to answer this question. If you did not answer the question, discuss the socio-economic effects of use of the death penalty in 14th century Europe.
6. Consider the horizontal cross section of an arrowhead, two sides of which are 4 cm long and the other 2 cm. Fixed perpendicularly to and at the middle of the shorter side is a shaft of uniform cross sectional width and length 40 cm. The mass of the arrowhead is equal to the mass of the shaft. Assuming masses are proportional to cross sectional areas, find without drawing a diagram
   1. the distance from centre of the shaft to the centre of mass of the system;
   2. the magnitude of the sum of the moments about this point assuming the arrowhead were of mass 0.1 kg and the shaft were
      1. parallel to the horizontal;
      2. and inclined at 15 degrees to the horizontal.
7. Robin Hood has laid an ambush for the Sheriff of Nottingham. He comes upon one of the said Sheriff's evil henchmen preparing to shoot a warning note into the town. The note is fired at 50 m/s at an angle of 60 degrees to the horizontal from the henchman's bow. If Robin Hood can fire an arrow at 70 m/s at 45 degrees to the horizontal from the same position as the evil henchman's, find
   1. an expression in i and j for the displacement of
      1. the evil henchman's arrow in terms of the time p after its launch;
      2. Robin Hood's arrow in terms of the time q after its launch;
   2. how long Robin Hood has to dispose of the evil henchman and launch his arrow if he wants to shoot the first arrow down;
   3. and, if Robin Hood were to slightly reduce the speed with which he launched his arrow, whether he would have slightly more or slightly less time to dispose of the evil henchman.
8. A horse is pulling a cart containing 50 kg of the Sheriff's gold up a slope inclined at 30 degrees to the horizontal. As the horse was eating a four leafed clover, one of the merry men shot an arrow into a money bag at back of the cart, allowing gold to dribble out. The initial mass of the cart, driver and gold is 300 kg. The driver, of mass 125.5 kg, whips up the horse, and immediately falls off. The horse only achieves its full power when the gold has all run out. At the moment in which the last of the gold falls from the cart, the instantaneous velocity of the cart up the slope was 1 m/s and its instantaneous acceleration 1 m/s/s up the slope. Find
   1. the ethnic origin of the horse;
   2. and an expression for the angle at which the slope would be inclined for the other data given in the question to remain constant, if a beast of burden exerting a power P on the cart were substituted.
9. A peasant draws a bucket of water from a well by pulling down on a frictionless pulley. The rope would normally be on the point of breaking if a mass of 20 kg were raised at 5 m/s/s. However, one of the Sheriff's evil henchmen has weakened the rope so that it will break if anything more than half this tension is applied. The distance to the bottom of the well is 10 m. Assuming the mass of the bucket is negligible, and modelling the rope as an inextensible light string and the peasant as a particle (if necessary) find
   1. the greatest acceleration with which 10 kg of water can be raised without the rope breaking;
   2. and the mass of water and acceleration of the raising of the bucket if the amount of water received by the peasant divided by the time of that raising is to be maximised, justifying any assumptions you make.
   Comment on the ethical implications of modeling the peasant as a particle.