1 Newton’s Laws Review • Newtonian mechanics is incredibly powerful; it applies to the vast majority of human- scale (and even interplanetary-scale) physics. • Apart from relativistic e ff ects at very high velocities (special relativity) or in very strong gravitational fields (general relativity), or incredibly small (atomic) length scales (quantum mechanics), Newton’s laws accurately describes the dynamics of our world. • Now on to Newton’s Three Laws (I write them as N1, N2, N3 for short) 2 Newton’s 1st Law N1: An object in motion remains in motion and an object at rest remains at rest unless acted upon by an outside force. – Called the "law of inertia", it just says that accelerations (changes in velocity) are brought about by forces (pushes or pulls). – History: Aristotle > Galileo > Newton... but really lots of others (e.g. Jean Buridan said this in like 1330) – Holds in an inertial reference frame (i.e. a non-accelerating reference frame). ∗ Let’s say you’re driving, you stop at a red light, and your co ff ee slides o ff of the center counsel. You won’t observe a force in your reference frame, that’s because your in an accelerating reference frame. Someone standing on the sidewalk will see that there was an external force on the car that slowed it down, that there was no such force acting on the co ff ee, therefore your co ff ee remained in motion while your car accelerated, and reduced it’s velocity to zero. ∗ Similarly for spinning reference frames like the Earth (if you want to get really technical). You might have heard of the Coriolis e ff ect, that’s the result of the rotation of the Earth. ∗ Now, if we’re talking about a stationary reference frame located on the Earth, often the rotation of the Earth is completely negligible. The assumption that this room is an inertial reference frame is just fine for any experiment I can do in my basement. – Outside force is key. Think about a system that involves a pool table with a bunch of billiard balls. If two balls collide that’s a force, but it doesn’t bring about an acceleration on the system (the pool table doesn’t move), why not, because it’s an internal force, it’s not a force being applied externally to the system. Now if the system you want to consider is instead a single pool ball, say it’s the 8 ball, then that pool ball getting hitting by the cue ball brings about an external force on the system (that’s just the 8 ball), and this external force does bring about an acceleration. 3 Newton’s 2nd Law N2: ~ F Net = m ~ a = m ̈ ~ x – Ok, this law is going to be the heart of this class, and for many of you, this eqn is pretty much going to be your life (e.g. if you’re a Mechanical Engineer). – This formalism assumes constant mass. If the mass of some object changes in time (e.g. a rocket burning fuel), then it’s technically ~ F Net = m ̇ ~ p , where ~ p = m ~ v is the momentum. – SI unit of force is the Newton (N), where 1 N = 1kg m s 2 – We’re talking about translational motion here, no rotating objects until Chap 10 of your text. – Quick example. Baseball of mass 0 15 kg thrown from rest (0 m / s ) to 30 m / s over a distance of 0 3 m . What the average force have I applied to the ball? Well, let’s pretend it’s 1D (approximately true): F x = ma x = (0 15 kg ) a x v x ( t ) 2 = v x 2 0 + 2 a x ∆ x (30 m / s ) 2 = (0 m / s ) 2 + 2 a x (0 3 m ) a x = (30 m / s ) 2 − (0 m / s ) 2 2(0 3 m ) = 1500 m s 2 F x = ma x = (0 15 kg )(1500 m s 2 ) = 225 N 4 Newton’s 3rd Law N3: ~ F 12 = − ~ F 21 – Sometimes stated "for every action, there is an equal and opposite reaction". Better to say that if there is a force exterted on body 1 by body 2, then there is also a force exerted on body 2 by body 1 that is equal in magnitude and opposite in direction. – Note the order of the indicies ~ F 12 refers to the force ON 1 BY 2. – Hammer sits on the board, we have ~ F hammer board = − ~ F board hammer ...that’s kinda obvi- ous and easy to believe. – Hammer hits board and it accelerates, still have ~ F hammer board = − ~ F board hammer ...that’s a little more surprising – What’s even more suprising is that this is still the case for non-contact forces like gravity. ~ F me Earth = − ~ F Earth me – Newton’s third law is really helpful if you’re standing in the middle of a frozen fric- tionless pond...just exert a force on something (throw your boot out in one direction) it’ll exert an equal and opposite force on you and you’ll slide your way to safety. 7 BD ACL 61 AN AJO BBVWS THR T l. asm \JJÜr-Sf-t7 (NOT È) flPPlSb s F6QCEs P€RAcPDzcoz..< Active Learning - C hap 4 - Dynamics: Newton ’ s Laws of Motion 28. (I) Draw the free - body diagram for a basketball player ( a ) just befo re leaving the ground on a jump, and ( b ) while in the air. See Fig. 4 – 34. 27 (I) A box weighing 77.0 N rests on a table. A rope tied to the box runs vertically upward over a pulley and a weight is hung from the other end (Fig. 4 – 33). Determine the force that the table exerts on the box if the weight hanging on the other side of t he pulley weighs ( a ) 30.0 N, ( b ) 60.0 N, and ( c ) 90.0 N. Η. CoUNT τ Gxv6M:F36 W ΥΌΙΔ γη (A) HAU6 λΙΕΧΤ (Κεξ? Ι.)ΝΙΤ9 (ooRD CASES гЕтв ̄ к) NET, ш = ЗОГ%Ј (А) б 2ND: =? ю- т Grv6fJ:F9B ктв ̄ NET, л.јЕтгИЈ 2ND: =? аж: -F2B+Fgw а: ы-Езв GJVEI\)., FIND: RB FRB