1. PH 201 Dr. Cecilia Vogel Lecture 17

2. REVIEW  CM OUTLINE  CM with holes  Thrust

3. 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 cut out of the hole  xi = where cm would be if there were no hole, xh = center of mass of hole treat a mass taken away as negative mass added!

4. 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

5. Expelling Gas  Consider one moment in time:  the rocket and its current load of fuel  mass m  velocity v  In a short time,  t, a small amount of exhaust gas is expelled mm  moves at speed v rel RELATIVE to the rocket,

6. Force on Gas  What force does the rocket have to exert on the gas to expel it  Use: Force = ma  mass =  m  ave acceleration =  v /  t   v = v f - v i = -v rel -0  so a ave = -v rel /  t  Force exerted by rocket on expelled gas  ave force = -  mv rel /  t  OR instantaneous force = -(dm/dt) v rel

7. 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  where v rel = speed of exhaust gas relative to rocket and d m /d t is rate at which mass is expelled (mass per unit time)

8. 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,  F=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)

9. 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,  as m decreases, a increases

10. Velocity of Rocket  Acceleration is decreasing  If thrust is the only force, then a=T/m  velocity does not increase linearly  it increases logarithmically  where m 0 and v 0 are initial values  and m and v are values at a later time

11. 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.