SOLVING NEWTON'S 2ND LAW PROBLEMS

IN A NUTSHELL 

And yes, THAT is to stress that how well you do on the rest of the course depends A LOT on how well you do on this part ;-)

Not that the rest of the course won't present some difficulties too. But if you can't do THIS, you'll be in trouble for everything else that's coming afterwards as well. If you get it though, you'll be in good shape.

And in my already-long experience, students find this part tough. IMHO, that's because they try and handle it the same way as they've handled everything else so far - understandably.  But it won't work here. You CANNOT think of this as an exercise in "plugging in", and then doing algebra. Not if you want to pass the course that is.

So brace yourself: this is by far the longest tutorial I'll put online, and by far the most detailed, because it's by far the most important one for this course.

How do I know the problem I'm trying to solve has to do with Newton's 2nd law?

If the problem talks about forces, you can bet it does (conceivably it might involve work instead, but in doubt bet on Newton's 2nd law!). Newton's 2nd law is the principle that tells you how the movement of an object changes (how the object accelerates), when they are forces acting on that object. Now, they are always forces acting on an object. But not all problems tell you or ask you anything about them. When a problem does that though... you're on the right page to deal with it!

If on top of that you're busy studying the Newton's 2nd law chapter... that you're going to need Newton's second law is a pretty safe bet!!!

Ok, so I think my problem involves Newton's second law. Now what?

DO figure out:

•  what object the problem is all about. Often that's obvious, but some problems make this their difficulty, if you're dealing with several objects for instance. 

• in what situation this object is: what it's doing or/and what is being done to it

DO NOT look at the question itself. Yes, I'm dead serious. Sure, you'll have to care enough to answer it, at some point. But that's only the final, glorious step. Kind of like climbing on the podium to receive the medal. But the race will have been won or lost long before that. So forget about it for now, or it will just confuse you. You'd just be asking yourself "how do I go about finding this" or "how do I go about answering that?" You don't need to ask yourself that, because you already know. Whatever they're asking, what you have to do first is apply Newton's 2nd law.

So, what's Newton's 2nd law, and what does it mean to "apply" it?

Newton's 2nd law is a Triumph of the Human Mind. It's the Prized Jewel that put an end to millenia of confusion about how and why objects move the way they move. It's the Beating Heart of Mechanics, and Mechanics was the First True Science... (not really, it could easily be argued that astronomy and optics beat it to the title). 

You get the idea, but cheesiness aside, that's as true as it can be of anything.

What's so special about Newton's 2nd law? Well nature can be deceiving sometimes. It sure looks to us like if something is not being pushed nor pulled, it just doesn't move. The snag is, in fact, everything's always being pushed or pulled by something, on planets at least. As far as classical mechanics is concerned, every object solid or not is being pulled downwards by the Earth. Yet most of what you see around you is just standing still. That's because there are other forces that stops that pull from making the object fall: overall they push in the opposite direction. So the first take home message is that in order to find out why an object is moving the way it does (or just standing still), you have to take into account all the forces on it. That's the "left hand side" of Newton's equation.

There's something else that's deceiving too: in almost all situations we encounter in everyday life, when we push something, we have to keep pushing if we want the object to keep moving. So our instincts tell us that forces make objects move as opposed to them just sitting there. That's not right though, and the way to fix our intuition is to think about this: what happens if you push on object on ice? It's going to keep going at the same speed, pretty much. Objects don't behave that way on most surfaces because most surfaces exert a force on them, that pushes opposite our own push. Objects don't stop moving because there's no force on them, they stop moving because there's a stop causing the stopping (usually, the frictional force). So the second take home message is this: forces try to accelerate an object, to change its velocity (its speed, or the direction in which it's moving). They don't simply try to keep it moving at some constant speed. That's the second take home message, and the "right hand side" of Newton's equation.

So now that you know what it means, here's what Newton's second law looks like:

Adding all the forces                 

that are acting on an object  (Σ F)    =  the mass of the object (m)  X  the acceleration it is moving at (a)

Roughly, the mass is a measure of how much "stuff", how much matter, the object is made of; how heavy it would feel if you had to carry it. What mass does in that equation is tell you that if you want to accelerate an elephant, even on ice, you're going to have to push a lot harder than if you're trying to accelerate a French poodle.

As soon as you realize that a problem involves Newton's second law, and have decided what object the problem is about you must:

1) Write down Newton's second law in symbols:

                                                                 Σ F→ = m . a→

2) Draw a diagram indicating all the forces that act on this object, what type of forces they are (using symbols, like W for weight), and their direction: i.e. a free body diagram.

The purpose of doing this is not to show that you know what forces there are (although that's nice). It is so you can replace of Σ F→by the correct expression (made up of lots of symbols!).

Σ F→ just means "the addition of all the forces that act on the object". That's very general, which is what makes it gorgeous: Σ F→ = m . a→ is true for any object you can think of, from here to the furthest confines of the Universe, now and as far back as you've ever heard of and for times long after what you can imagine. It's true for planets and suns, elephant and microbes, you and I, that book on your desk and that car that the problem in that book is asking you something about. To have just one single, tiny equation like Σ F→ = m . a→ that is true for all that is amazing.

It's also pretty useless if that's all you've got, because surely that car in your problem isn't doing the same thing as a figure skater. You've got to adapt Σ F→ = m . a→ to the situation you're dealing with. Σ F→ = m . a→ is like an incredibly compressed file that has all this information in it, but at some point you've got to unzip that file. You've got to go from this unbelievably general statement, to something specific to what you happen to be caring about. Concretely, that means you have to figure out what Σ F→ is in the situation the problem is about. 

Doing this involves two steps:

• 1) writing down Σ F→ in diagram form (that's called a free body diagram)

• 2) Using 1), writing down Σ F→ in algebraic form (using symbols).

It's not a matter of picking between the two. You need to do BOTH, in that order.

"FREE BODY" DIAGRAM (DIAGRAM OF FORCES)

IN A NUTSHELL

A free body diagram is NOT a drawing. It is a part of solving the problem exactly like writing down an equation is.

DO NOT draw the object. Represent it as simply as possible, by a point or a square.

The forces should look like they "radiate" from the object: they should start on it.Like this for example:

Ok, but how do I find the forces?

Look for what could be causing them:

1) if the object is near the Earth - or some other planet, or star, etc... (i.e. pretty much everything but The Enterprise), weight W is acting on it. (Weight is a force in physics, it's not how many kilograms of the stuff there are).

direction: straight down, towards the center of the Earth or other planet. 

magnitude: mass of the object X acceleration due to gravity, that is mg (and on Earth g is 9.8 m/s²).
Warning! mg is just the magnitude of the weight that acts on the object. It absolutely does not mean that the object is really accelerating at g!!! What the object does depends on ALL the forces that act on it. It only accelerates at g if the weight is the only force acting on it (i.e. if it's falling!).

2) if there is a rope, string, or spring pulling on the object, tension T is acting on the object (one per string, etc...).

direction: along the rope (string, etc...), away from the object. 

magnitude: there's no general expression for it (the problem will usually tell you what it is or ask you to find it).

3) if the object is in contact with a surface:

• there's a normal force N acting on the object.
  direction: perpendicular to the surface (that's what normal means), away from the object. 
  magnitude: there's no general expression for it (the problem will usually involve you having to find it).

• chances are there may be a frictional force, too: 
• if the object is rubbing against the surface there's a kinetic frictional force f_k.
  direction: along the surface, in the direction opposite the movement of the object. 
  magnitude:  μ_k N.
• if the surface is preventing the object from slipping, there's a static frictional force f_s.
  direction: along the surface, in the direction opposite that the object "wants" to slip. 
  magnitude:  Usually unkwown. If the object is just about to slip, then it is μ_s N. Otherwise, it is less than that, but you don't know how much less (the static friction force prevents the object from slipping; if it's far from slipping, it doesn't take as much force from the surface to stop it).

4) the problem may tell you that there is someone/something pulling or pushing on your object, too.

Write the symbol for the forces on the free body diagram (W, N, T, etc...). If there are several forces of the same kind, use subscripts to tell them apart (say, T₁ and T₂).

There exist other forces as well (air resistance, usually neglected; buoyant force, lift, etc...), but the above are those that you will most often encounter in problems.

FROM THE FREE BODY DIAGRAM TO NEWTON'S 2ND LAW

IN A NUTSHELL

 
Ok, I've drawn the free-body diagram for the object I'm interested in. Now What?

1) Now choose which directions to call "x" and "y", by drawing something like this near your free body diagram (for example):

This means the positive x-direction is to the right and the positive y-direction to the left.
Another example would be this:

There is no right and wrong choice here, you are just choosing a convention. However.
1) A choice does need to be made!
2) Some choices are sometimes more convenient than others. If your object is accelerating, pick the direction of its acceleration as your x (or y) direction. If it's not accelerating, pick axes in such a way that the most forces are going to point along some of these axes. It will make the maths easier (this is why people choose tilted axes when dealing with an object on an inclined plane).

Right, I've got the free body diagram AND the axes. Now what?

Remember you want to apply Newton second law, that is you want to find out what it is in the situation you're dealing with. In its most general form, Newton's 2nd law is this:


                                                                Σ F→ = m . a→


That the F and the a have arrows over them mean that they're vectors. Concretely, this means that this equation is true in ALL directions. That is, if you add up all that the forces do in the horizontal direction, that's going to be equal to the mass of the object times how the object in the horizontal direction. The same goes for the vertical direction, and up-down, and... any direction you can think of. For an infinite number of directions. To solve the problem that's way more equations than you need. You need as many equations as the problem has "dimensions": if everything is pulling/pushing along just one line, one equation will do. Usually in the problem's you're given, you have forces pulling/pushing in two dimensions, all in a plane, so you need two equations. That's why you chose two axes above.

So:

2) Write down Newton's 2nd law for each direction you have defined (usually x and y). That is, if you are dealing with a problem in two dimensions, write:


                   Σ F_x = m . a_x                                                            Σ F_y = m . a_y

Notice: the arrows are gone. The Fs and as in there aren't vectors any more. They are symbols that represent components. 

Here comes the big step:

3) Write them down again, but now replace Σ F_x and Σ F_y by what these sums actually are in the specific case you're dealing with. This is what you drew the free body diagram for. This is when you write Σ F→ (which has now become Σ F_x and Σ F_y ) in algebraic form.

How do you do that? Well the only way to explain it clearly is to use an example, so here goes.

Say you've got this free body diagram and you've picked for itfor it the x and y axes below :

and of course you've written this:         Σ F_x = m . a_x                                                            Σ F_y = m . a_y

Let's start with Σ F_x = m . a_x :

a) Ask yourself: which of these forces contribute to pulling/pushing the object along the x direction (whether it's towards + x or -x )?

Well T₁ does; W doesn't; and T₂ does. So below Σ F_x = m . a_x write:

Σ F_x  = m . a_x

   T₁                   T₂                    = m . a_x

Notice I've left lot's of space: that's because I know that more things are likely to go in.

b) Now ask yourself: are these forces pulling/pushing the object towards + x or towards - x? 

If a force tends to pull/push towards + x, what goes in front of it is a +, if it tends to pull/push towards - x, put a - sign in front of it (on the same line ;-).

In this case: T₁ tends to pull towards - x, and T₂ towards + x. So we get:

  - T_{1                 +}  T₂                    = m . a_x

Note that there's nothing mysterious about it, it's all as intuitive is it could possibly get: if you've got two forces that tend to pull something in opposite directions, the object is going to react as if a smaller force was pulling it. To take a simpler example than the one we've got here, in the picture at the start of this section, whether the nut's going to accelerate towards Scrat only if he's pulling on it stronger than his girlfriend does. If they're both pulling as strong, the nut's not going to accelerate towards either of them (the sum of their forces on the nut, the net force on the nut, will be 0). Now sure, if you're being pulled hard in different directions you're not going to feel the same thing as if you're being pulled gently in only one: you can feel yourself being pulled apart. That's true but in both cases you'll accelerate the same amount in the same direction, and that's what we care about here, that's what Newton's second law is about.

But back to our case. Notice I still left space between the symbols. Because we're not done yet.

c) Ask yourself: for each force, is the entire force pulling/pushing along x, or is it just part of it... just a component of it?

If it's the whole force, you're done with that one. If it's not, you're in for some trig. 

Trig happens all the time in these problems, so hoping you get lucky enough not to have to deal with it too much is not an option. The only Path to a Happy and Stress-free Life here is to get good at it.

In fact what you're writing down here are components of forces.The component of a force always has the structure:

magnitude of the force (like, say, T₁)      .        either cos(angle) or sin(angle)

That makes sense, too: only part of the force does the pulling/pushing in that direction. So what has to end up in your equation must be smaller than the whole force, than it's entire magnitude. So you need to reduce that magnitude, and cos(angle) or sin(angle), because they're less than 1, achieve that very well! (This doesn't mean that we picked cos and sin to do the job, but could have chosen something else. Nature happens to have picked cos and sin. And we're really very lucky that it did: it means that forces behave like lengths in a way, that the relation between how strongly a force pulls in a certain direction compared to how strong the force is, is the same relation as how much, say, a stick goes in a certain direction, compared to how long the stick is. That's really as simple as we could have hoped!)

Ok, so let's do some trig. First let's zoom in on T₁.

We're given α, so the question is: what is going to give us how much of T₁ is along x, cos α or sin α?

Imagine a triangle with T1 the hypotenuse, and with α at one of the corners. Meaning, this:

Is α adjacent to or opposite the x direction? Well it's adjacent to it. So pick cos α. Now you have:

-  T₁  cos α +   T₂                    = m . a_x

Now let's zoom in on T₂.

Imagine the triangle with T₂ as one of the sides and that contains β. In this triangle, picture the side that's parallel to the x axis. It would be the one in red:

Is this side adjacent or opposite to β? Well it's opposite. So pick sin β. Your completed equation is:

- T₁ cos α + T₂ sin β  = m a_x

Note that you cannot always use cos when dealing with x-components: it depends on which angle you are given. 

Also note this: nearly always, you will be given/expected to use angles between the force and the nearest axis direction. These angles are less than 90^o, so when you take the cos or sin of them, you always end up with a positive number. This is why you have to impose the sign yourself in step b): it is not automatically going to pop out as if you were counting angles from the positive x-direction. So you have to ensure yourself that the sign of each term (that is, each expression like -  T₁  cos α , or like  T₂  sin β ) is positive when it makes physical sense for it to be, ad negative otherwise. Physical sense here means: does it tend to pull the object in the positive or the negative x direction?

                You have just written Newton's second law for your object, in the x-direction!

Now you need to do the same thing in the y-direction. Scroll back to the free-body diagram and try to do it on your own. When you are done compare to the answer below. You should find this:

T₁ sin α + T₂ cos β - W = m a_y

If you want to have an ok time in this course, practice doing this until you can do it in your sleep!




TO SUM UP: You entire problem so far should look like this:

Σ F_x = m . a_x                                                            Σ F_y = m . a_y

-  T₁  cos α + T₂  sin β                                T= m . a_x  1sin α + T₂ cos β - W = m . a_y

That's all.... one diagram, a pair of axes and two lines!!! But they've got to be correct!

Once you've done that in a problem, you're practically done. Go back to the text of the problem and re-read it. Then:

4) if you can, simplify the right hand side of the equations: ask yourself if according to the problem, a_x or  a_y is 0. If, say, a_y is 0, well set it to 0. That also means that the entire acceleration is in the x direction, so a_x is just a itself. In this case you would end up with:

                -  T₁  cos α + T₂  sin β                                T= m . a  1sin α + T₂ cos β - W = 0.

5) check what question the problem was actually asking! Now is the time to start worrying about it. In problems involving only Newton's second law, it will be a matter of doing algebra on the two equations that you've just obtained.

THAT'S ALL FOLKS!