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PH 201 Dr. Cecilia Vogel Lecture 17 REVIEW CM OUTLINE CM with holes Thrust CM of object with hole How do you find the center of mass of an object with a hole – if you know where its cm would be if it didn’t have the hole? let mi=mass before hole, mf = mass after it has a hole, mh = mass of stuff x cut out of the hole xi = where cm would be if there were no hole, xh = center of mass of hole CM xCM mi xi mh xh mi mh treat a mass taken away as negative mass added! Recall In Car Talk puzzler, man throws a toolkit he moves the opposite direction the CM of the two keeps moving the way it was This is how jets and rockets work: exhaust gases are ejected backward, and rocket goes forward Thrust is the force the gases exert on the rocket Expelling Gas Consider one moment in time: the rocket and its current load of fuel mass m velocity v In a short time, Dt, a small amount of exhaust gas is expelled Dm moves at speed vrel RELATIVE to the rocket, Force on Gas What force does the rocket have to exert on the gas to expel it Use: Force = ma mass = Dm ave acceleration = Dv /Dt Dv = vf - vi = -vrel-0 so aave = -vrel/Dt Force exerted by rocket on expelled gas ave force = - Dmvrel/ Dt OR instantaneous force = -(dm/dt) vrel Thrust But the thrust is the force exerted by the gas on the rocket. this is the reaction force equal and opposite The thrust force on the rocket is dm T vrel dt where vrel = speed of exhaust gas relative to rocket and dm/dt is rate at which mass is expelled (mass per unit time) Please Note: This applies not only to rockets expelling exhaust gases, but also to anything continuously losing or gaining mass Equation in text is misleading T = ma is only true if T is the only force! Generally, SF ma SF=ma and T is one of the forces you sum Direction of thrust is always opposite direction of mass expelled (or in the direction of mass impinging) Varying Acceleration Can you have varying acceleration with constant force?? if m variese like a rocket mass is lost at a rate of dm/dt and mass is decreasing Suppose constant thrust is the only force, T m( t ) a ( t ) as m decreases, a increases Velocity of Rocket Acceleration is decreasing If thrust is the only force, then a=T/m velocity does not increase linearly it increases logarithmically m0 v v0 vrel ln m where m0 and v0 are initial values and m and v are values at a later time Thrust Example At one instant in time, a balloon filled with air has a mass of 3.1 g. The balloon is moving at 1.1 m/s and expelling air at a rate of 150 cm3/s at that instant. The expelled air travels at a speed of 2.7 m/s relative to the balloon. Pretend there is no drag. Find the speed of the balloon 0.25 s later, if the balloon continues to expel air at approximately constant rate and speed during that time.