Mathematical Tripos Part IA Michaelmas term2011 Mechanics(non-examinable)Exercises for lectures7and8Dr S.T.C.Siklos Comments and corrections:e-mail to stcs@cam.All examples sheets and solutions are avail-able on www.amtp.cam.ac.uk/user/stcs/mechanics.html
On these sheets,no attempt is made to‘model’real-life situations:no trains,cars,cyclists, lifts,governors of steam engines,etc.It is assumed that there are no‘real’forces,such as air-resistance unless they are specifically mentioned.Most questions,but not all,avoid numbers and units,prefering general algebraic formulae with consistent dimensions.
1A particle is released from rest and falls under the action of gravity.By calculating its speed after falling a distance h,verify that the principle of conservation of energy holds in this situation.
2A particle of mass m is projected with initial speed u up a line of maximum slope of a rough plane inclined atαto the horizontal.The coefficient of friction isµ(assume the frictional force isµ×normalreaction).Show that the acceleration down the plane is g(sinα+µcosα) and hence calculate the distance s up the plane at which the particle comes to rest.Use the conservation of energy to show that the amount of work done against the frictional force is
1 2mu2
µcosα
sinα+µcosα
.
Verify this result using work done against frictional force=frictional force×distance moved.
3Use the principle of conservation of energy carefully!tofind the maximum height of a particle projected with speed V at an angle ofαto the horizontal,noting that the horizontal velocity is constant.
4A light inelastic string passes over two small smooth pulleys A and B at the same horizontal level a distance2a apart.Particles of mass m are attached to either end and a particle of mass M(where M<2m)is attached to the midpoint.The system is released from rest with the particle of mass M at the midpoint of AB.Using conservation of energy,show that the system next comes to rest when the particle of mass M has fallen a distance
4amM
4m2−M2
.
pulleys5Water is pumped to the surface of the Earth from a depth d and issues from a pipe of cross-sectional area A at a speed of v.The density of the water isρ.Using energy considerations,find the power of the pump.
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