Newton s second law: dp dt. p = mv. Sergi Hildebrandt Cahill 364)

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1 Newton s second law: SF dp dt S p = mv Sergi Hildebrandt (srh@caltech.edu. Cahill 364)

2 Watch also: Video 1 Video 2

4 F M B µs M A 0 µ s = 0 For M A, M B and µ s, find the maximum force, F, that can be applied without sliding M B

The most interesting thing to learn from this problem is the action/reaction pair due to friction between body A and B and their direction (to the right in B and to the left in A).

5 The most interesting thing to learn from this problem is the action/reaction pair due to friction between body A and B and their direction (to the right in B and to the left in A). F f M B µs M A f 0 µ s = 0 f = M B a B and F f = M A a A We want to impose a A =a B =a. Summing up get: F = (M A +M B ) a. On the other hand, the static friction is at most f=µ s N B =µ s M B g. Therefore, the maximum acceleration is a=f/m B = µ s g and: F max =(M A +M B )a max = (M A +M B )µ s g

6 FP9 from the Course webpage

13 e) Not yet. Later on in the course, when we deal with angular momentum

14 Consider a flexible chain of length L and linear density l (M=lL)lying on an incline as shown in the figure. There s friction all along the incline. The chain is initially at x=x 0 and at rest. x a b 1) Find the acceleration of the chain in terms of x and any other relevant variable of the problem 2) Find the speed when x=0

21 Write it down!

22 Write it down! F = m a?

23 Write it down! F = m a? NOPE!

24 Write it down! F = d(mv)/dt!

25 F = d(mv)/dt F = v dm/dt + ma!

26 If m = constant, then F = ma

27 F = ma Who wrote this equation? Probably the most used equation in physics!

28 ?

29 1707, Basel, Switzerland 1783, Saint Petersburg, Russia Leonhard Euler 65 years after Newton published it!

30 What s the difference compared to the problem of the chain on an incline? Lifting up a chain Chain falling through a hole

31 y

37 y

DR. NO S ANTIGRAVITY MACHINE SERGI R HILDEBRANDT Figure 1 Dr. No wants to get big money from some CEO and Army Generals by selling the idea that he has discovered the secret for antigrativity.

44 DR. NO S ANTIGRAVITY MACHINE SERGI R HILDEBRANDT Figure 1 Dr. No wants to get big money from some CEO and Army Generals by selling the idea that he has discovered the secret for antigrativity. He plans to launch his Dr.No-Box1 and once it has attained a certain speeed, v 0,he llturno the engines as he pushes the button of what he claims is the antigravity engine. He will show to his audience that the spacecraft will continue to move upwards with constant velocity, despite the gravitational attraction of the Earth. After some demonstration time, he ll turn o the antigravity engine and come back to the ground, landing with some parachutes. Please, answer the following questions: a) How is it possible? b) The picture on the right hand side, shows what his fooling plan looks like: he will eject mass from the original spacecraft, after he pushes the antigravity button and during the demonstration time. The mass and speed with which the mass is ejected on both sides of the spacecraft is the same, so there is no horizontal force or motion. Find the relationship between v 0, g (the Earth s 1

45 2 SERGI R HILDEBRANDT gravitational acceleration), the initial mass of the whole system, M 0 and the time of the antigravity demonstration, t. c) How much mass should be ejected if v 0 =10m/s, t =1min?Doesitmakesenseinpractice? (Hint: assume that the minimum mass at the end has to be the mass of the enclosure of the spacecraft, plus engines, people, etc... say about 10,000 kg). d) Dr. No s engines cannot speed up Dr.No-Box1 faster than the speed of sound (he also wants to keep his tests silent ). What would be the minimum initial weight of the spacecraft be if the antigravity demonstration is a minute long?

46 DR. NO S ANTIGRAVITY MACHINE 3 1. Solutions a) Yes. Force is equal to d(m~v)/dt, sothatvelocitymaybeconstantifthemassvaries. The spacecraft can only lose mass. Therefore, dm/dt < 0andthatimpliesthatthedirectionofthe velocity has to be the contrary of that of the force. The trick may work when moving upwards, but not when descending.

47 4 SERGI R HILDEBRANDT 2. Solutions a) Yes. Force is equal to d(m~v)/dt, sothatvelocitymaybeconstantifthemassvaries. The spacecraft can only lose mass. Therefore, dm/dt < 0andthatimpliesthatthedirectionofthe velocity has to be the contrary of that of the force. The trick may work when moving upwards, but not when descending. b) F ~ = ~v 0 dm/dt, with~v 0 the velocity when Dr. No stops the engines of his spacecraft. On the other hand, W ~ = m~g (~g is pointing downwards). Consequently, i) ~v and ~g must be parallel. One can leave the sign of the direction free and check that + would imply increase of mass, and decrease. Let s choose, as explained in a) the negative sign and re-arrange W ~ = ~v 0 dm/dt: g/v 0 dt = dm/m, which is a simple di erential equation. Integrating, with the initial condition of m(t = t 0 )=m 0 : (1) m(t) =m 0 e g/v 0 t, ~v 0 = (v 0 /g)~g = v 0ˆk, where t is the time of the demonstration ( = t f t 0,ifonewishes). Itmakessensedimensionally, and ~ W = d(m~v 0 )/dt is satisfied.

48 DR. NO S ANTIGRAVITY MACHINE 5 3. Solutions a) Yes. Force is equal to d(m~v)/dt, sothatvelocitymaybeconstantifthemassvaries. The spacecraft can only lose mass. Therefore, dm/dt < 0andthatimpliesthatthedirectionofthe velocity has to be the contrary of that of the force. The trick may work when moving upwards, but not when descending. b) F ~ = ~v 0 dm/dt, with~v 0 the velocity when Dr. No stops the engines of his spacecraft. On the other hand, W ~ = m~g (~g is pointing downwards). Consequently, i) ~v and ~g must be parallel. One can leave the sign of the direction free and check that + would imply increase of mass, and decrease. Let s choose, as explained in a) the negative sign and re-arrange W ~ = ~v 0 dm/dt: g/v 0 dt = dm/m, which is a simple di erential equation. Integrating, with the initial condition of m(t = t 0 )=m 0 : (2) m(t) =m 0 e g/v 0 t, ~v 0 = (v 0 /g)~g = v 0ˆk, where t is the time of the demonstration ( = t f t 0,ifonewishes). Itmakessensedimensionally, and ~ W = d(m~v 0 )/dt is satisfied. c) The final mass will be: m F = m 0 e 58.8 = m 0!! No way in practice of having m 0.Recall m F =10, 000 kg. However: d) The speed of sound is approximately 342 m/s at sea level and at 20 C. Now, m F = m 0 e 1.72 = 0.18m 0. This sets the minimum mass of the spacecraft. Less will not provide enough mass to be ejected. If the final mass is 10,000 kg, then the minimum initial mass of Dr.No-Box1 has to be: m 0 =10, 000/0.18 = 55, 806 kg. Seems okay! Let s do it. address: srh@caltech.edu

Forces and Newton s Laws Reading Notes. Give an example of a force you have experienced continuously all your life.

Forces and Newton s Laws Reading Notes Name: Section 4-1: Force What is force? Give an example of a force you have experienced continuously all your life. Give an example of a situation where an object