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PES 1110 Fall 2013, Spendier Lecture 10/Page 1 Today: - Newton’s 2nd law - HW due today - Seminar announcement: Please join us for a very interesting seminar on a telescope that uses a thin plastic membrane as a lens. Called a photon sieve, it works by bending light through billions of tiny holes in the "cling film" style optic. Unlike in existing traditional mirror-based telescopes, the new lens can be folded to fit into a tiny space. 11am, Friday September 20, Room A204 Osborne - Will work hard to get practice exam out by Friday! - Friday we will have 2nd Quiz! Recap Monday: Forces to be identified in any problem: - Weight - w , the downward force on an object due to gravity. - Normal Force - Fn , the perpendicular force exerted by one solid object onto another solid object. - Friction - f , force which slows a moving object, always opposed to the motion, 180 degree away from velocity) opposite to v . - Tension - T , pulling force exerted by rope, chain, or spring, always at same angle as rope. If you have a rope at 30 degrees in a problem the tension vector will have the same angle. Lots of components of vectors and vector addition. More on weight: mass SI kg British slug English Engineering lbm (pound mass) a m/s2 ft/s2 ft/s2 force (weight) m kg/s2 = Newton slug ft/ s2 = pounds lbm ft/ s2 = lbf (pound force) How do we measure forces? We can use a spring balance: The distance x which the object moves due to the object's weight is proportional to the force. (We will learn later: F = kx, k is a constant describing the stiffness of the spring) PES 1110 Fall 2013, Spendier Lecture 10/Page 2 Newton’s First Law: The Law of Inertia An object at rest stays at rest, an object in uniform motion stays in uniform motion if (and only if) the net force acting on the object is zero. ∑F =0 net Uniform motion - Straight line and constant speed, i.e, constant velocity. Inertia - The property of all matter to stay in motion if already in motion; to stay at rest if already at rest. Inertial Reference Frames Reference frames in which Newtonian mechanics holds are called inertial reference frames or inertial frames. These are reference frames that move with constant velocity , including v = 0. We typically say that the ground is an inertial frame. But we know from last lecture that the earth’s rotation causes a small centripetal acceleration. We calculated it to be 0.3% of g and hence we will ignore it and treat the earth surface as an inertial frame. Said something stupid last time: a hanging object has two forces (tension and weight) acting on it – the only problem where there is only one force is a free-falling object! First Law Example 1) A 5 kg mass is hung from the ceiling using a "massless" rope. What is the magnitude of the tension force exerted by the rope on the mass? Hint: A 5 kg mass has a weight of 49N on earth where this problem is taking place. Answer: 49 N First Law Example 2) Two 5 kg masses are connected to each over pulleys using a rope. What is the tension force that the rope exerts on the right-hand mass if they are both at rest? Answer: 49 N PES 1110 Fall 2013, Spendier Lecture 10/Page 3 Newton’s Second Law: This is only the really equation we will be using in chapters 5 and 6. We will be coming back to this in a big way in chapter 6. Right now we will give a brief introduction and then practice hard using it. We will see that we don’t really need a first law, we talked about it for historical reasons, since the first law is contained in the second law. F ∑ = 0 then we have a constant velocity net dv Constant v = constant ⇒ =a=0 dt So if the forces do not add up to zero – what does it tell us? The first law tells us that if ∑F ≠0⇒a≠0 net Then the acceleration is also not zero. So the first law is the case of zero acceleration that the second law itself contains. So the big thing is that: Forces cause acceleration This is the physical effect – when the forces do not add up to zero the acceleration is nonzero. This is why we had to do kinematics first because it would not have made that much sense if you did not understand what acceleration is. This is how nature works, from the bottom up, forces cause acceleration, the acceleration in turn determines the velocity, and then acceleration and velocity together determine the position. We learned it the other way position-velocity-acceleration-forces Sir Newton found more than that. He found that the acceleration is: (a) In the same direction as the net force We learned that acceleration vectors are tricky things. They can change both speed and direction. So to help you out it is a good thing to remember that when an object starts from rest, the direction in which an object moves is in the direction of its acceleration. If I push an object from rest to the right the net force will be to the right and it will accelerate to the right. (b) (the amount of acceleration is) Directly proportional to the net force So I exert some number of Newton’s to the cart it accelerates. If I double the amount of force I exert, apply twice as much force, it will give me twice as much acceleration. (c) Inversely proportional to the mass I can push a block for you. Can I push the wall? I cannot make the wall accelerate. The wall is too big. How do we measure how big the wall is? We throw in the “mass” with the more mass the smaller the acceleration. Mass is the amount of matter inside an object. PES 1110 Fall 2013, Spendier Lecture 10/Page 4 It is the number of protons and neutrons inside of an object, that determine mass. More mass for a given force the smaller the acceleration. Putting these three things together you get: Second Law of Motion: ∑F net m =a ⇒ ∑ F = ma net kg i m units 2 = [ N ]...Newton s The net force is equal the mass times acceleration! (Don't say F = ma, here F stands for single force!) Here we stress that it is not just one force which determined the acceleration, it is all of the forces together, it is the net force. It is the vector addition of the forces which equals mass times acceleration. Be aware, to apply the law we have to use mass in kg. So in problems you need to make sure that the mass is given in kg – otherwise you need to convert! Second Law Examples Example 1: A 6860N car is in free-fall, what it its mass? An object in free-fall has the only force acting on it – it is gravity – this is the world famous single force problem! Object in free fall has acceleration 9.8 m/s2 w = m g (weight and acceleration are downward) m = W/g = 6860/9.8 = 700 kg Example 2: A 6860N car is sitting stationary on the ground, what is its mass? It will have the same mass – I will have the same number of atoms in it. PES 1110 Fall 2013, Spendier Lecture 10/Page 5 Example 3: Example: A 6860N car is traveling with a speed of 30m/s when the driver removes her foot from the gas pedal, making the engine force zero. If the frictional force is 350N, what is the car’s acceleration? PES 1110 Fall 2013, Spendier Lecture 10/Page 6 NOTE: If an object is already in motion, just because you push on it, it does not have to go in the direction you pushed! It will accelerate in the direction you push it. What about a car pushed from rest? If I push an object that is initially at rest, it will move to the left, the direction I pushed. Because the acceleration and velocity are in the same direction. In the car example above: If a car is already moving to the right and I push on it to the left, the push is causing an acceleration to the left, and an acceleration opposite to velocity will slow the object down. It causes a deceleration. An object already in motion will not move in the direction of the net force. It will accelerate in the direction of the net force. Example 4: A spaceship is floating sideways from point A to B in the middle of outer space. At B the spaceship turns on it’s engines for 5 s and moves to point C. At C the engines are again turned off and the spaceship floats to point D. Which of the following picture correctly shows the spaceship’s trajectory from A to D? Solution: From A to B rocket goes on straight line since no force ==> uniform motion! When rockets turned on this causes an acceleration in y-direction but NOT x-direction ==> acceleration motion in y, uniform motion in x ==> curved motion PES 1110 Fall 2013, Spendier Lecture 10/Page 7 When rockets turned off again we go back to uniform motion ==> straight line. But now we have velocity in both x- and y-directions ==> straight line at an angle From A to B rocket goes on straight line since no force ==> uniform motion! Example 5) Two forces are acting on an object and it is moving with constant velocity. One of the ⌢ forces in F1 = (3.0N) ˆi - (6.1N) j . What is the other force F2 ? Example 6 a) A bank robber is trying to push a safe on a frictionless floor. He can apply a horizontal force of 2.0N. If the safe's mass is 800 kg, what acceleration can he produce on the safe, if the safe starts from rest? PES 1110 Fall 2013, Spendier Lecture 10/Page 8 Example 6 b) If the bank robber above was sliding the safe on a floor with friction equal to 0.70 N, what would the acceleration of the safe be? Example 6 c) The bank robber above needs to push the safe 4.0m. If the safe is originally at rest, how long does it take? (The bank robber can accelerate the safe 1.625 x10-3 m/s2 in the horizontal direction of his push)