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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: xox.happy1.xox on October 15, 2008, 09:01:33 pm

Title: Help me understand Mechanics!
Post by: xox.happy1.xox on October 15, 2008, 09:01:33 pm: I am stuck! I at least have some grasp of the other stuff, but don't have a clue what mechanics is about! I don't like Newton (as I have expressed a million times!), and his laws with tension ropes on a pulley and mass at an inclined angle... I just don't get it :'( I can't even add the i, j and k vectors. Should I just give up on Mechanics, or can someone kindly explain this concept to me :)?
Title: Re: Help me understand Mechanics!
Post by: ice_blockie on October 15, 2008, 09:18:51 pm: I'll write a quick summary which I will post later...
Title: Re: Help me understand Mechanics!
Post by: xox.happy1.xox on October 15, 2008, 09:20:30 pm: Yay... Thanks! :)
Title: Re: Help me understand Mechanics!
Post by: ice_blockie on October 15, 2008, 09:48:08 pm: (http://img389.imageshack.us/img389/9527/47822908ny2.th.png)

Pretend we have a pulley with box of mass $m$ and mass $M$ hanging on either side. $(M > m)$

To find acceleration $a$ , and tension $T$ , we have to find the equations of motion (fancy name for what the hell is going on for each particle, sort of like predicting what the boxes are going to do)

Now we need to have a reference direction, like how we have ‘left’ and ‘right’ ‘up’ and ‘down’, we need the direction to understand…

Now $i$ and $j$ components will not help in this case since the boxes are going in different directions. What we need to find is the direction where both are going. Pretend you were holding the two box things, if you let go of both, the heavier one will fall. So we will say the direction of motion is the direction which they move: up and around the pulley and back down the side with the heavier box.

Now what we do is apply $R=ma$ (how the box moves = mass x acceleration) to each box individually. Pretend the other box does not exist for now.

Now we can get an equation (1) from looking at the diagram. So according to our reference direction what do the boxes and forces do? For the smaller mass, m, it goes up. So tension is ‘helping’ it go up, so $T$ is positive. Gravity is the opposite, trying to pull it down, so mg is negative. Using $R=ma$ , we get (1).

(1) $T -mg = ma$

Remember the reference direction is up around the pulley down the big mass. So how is the gravity affecting the big mass? It is helping it go so $mg$ is positive. Tension is opposing it so $T$ is negative. Adding these up gives mass x acceleration.

(2) $mg - T =ma$

To find a, we can add (1) and (2) like simultaneous equations, rearrange and blah which gives:

$a=\frac{(M-m)g}{M+m}$

To find $T$ , we can substitute $a$ back in (1) and then solve for $a$ .

We can apply the sample principle to all cases, remembering that we must find a reference direction which is the easiest for calculating things. If you don't understand any of this, just ask.
Title: Re: Help me understand Mechanics!
Post by: ice_blockie on October 15, 2008, 10:00:30 pm: I will do inclined planes....later...
Title: Re: Help me understand Mechanics!
Post by: fredrick on October 15, 2008, 10:29:21 pm: Suppose we have a mass m kg on a rough incline plane at an angle of 30 degrees to the horizontal, with a coefficient of friction of 0.4. Find the force P where the mass is on the point of moving up the plane(in terms of the other variables).
[IMG]http://img80.imageshack.us/img80/3023/mechem7.png[/img]
[IMG]http://img80.imageshack.us/img80/mechem7.png/1/w248.png[/img]

Firstly, we have a normal reaction force (N) which acts in a direction perpendicular to the incline.
We can resolve the forces acting perpendicular and we get:

$N-mgcos30=0$ , where mgcos30 is the perpendicular component of the force.
$N=mgcos30$

Now we have resolved the perpendicular forces and will move on the resolve the forces which act parallel (or antiparallel) to the incline.

Here will will assume going up the plane to be the positive direction.

Before we start with that you should note that the friction that acts in the opposite direction and has a magnitude of $Fr=\mu N$

Resolving in direction of incline we get:

$P-mgsin30-Fr=ma$

We can determine the friction $Fr=0.4mgcos30$

Hence $P-mgsin30-0.4mgcos30=0$ acceleration is 0 as it is on the point of moving.

Finally $P=mg(0.4cos30+sin30)$
$P=mg(\frac{3^\frac{1}{2}}{5}+\frac{1}{2})$
Title: Re: Help me understand Mechanics!
Post by: xox.happy1.xox on October 16, 2008, 09:08:46 pm: Can I just memorise these formulas as they apply to all cases? I don't think I will ever understand this... But thanks for everyone's help! :)

And ice_blockie, can you please do the working out for how you actually got the acceleration formula for pulley forces... I am so slow!
Title: Re: Help me understand Mechanics!
Post by: bucket on October 16, 2008, 09:22:33 pm: wowwwww i wish i did specialist, thats so cool!
Title: Re: Help me understand Mechanics!
Post by: xox.happy1.xox on October 16, 2008, 09:36:25 pm: No you don't. :P Once it is on exams and you don't know what you are doing... But on the other hand... It scales well. :)
Title: Re: Help me understand Mechanics!
Post by: Mao on October 16, 2008, 09:50:08 pm: the two equations are derived as following:

the system is moving in the direction towards the heavier mass [at constant acceleration]

the heavier mass is moving downwards, being pulled down by gravity (Mg) and up by T. It has a net acceleration of a downwards, so making downwards positive:

Ma = Mg - T

the lighter mass is moving upwards, being pulled up by T and down by gravity mg. It also has a net acceleration of a, but is upwards. making upwards positive:

ma = T - mg
Title: Re: Help me understand Mechanics!
Post by: ice_blockie on October 16, 2008, 10:13:47 pm: I hope you mean how you go from these

$Ma = Mg - T$ (1)

$ma = T - mg$ (2)

to that big ugly thing for $a= \frac{(M-m)g}{a(M+m)}$

Add LHS of both = RHS of both

$Ma + ma = Mg - T + T - mg$ <---WOW the 't's can cancel!
$Ma + ma = Mg - mg$ <--- t's are gone since $-T+T=0$
$a(M+m) = Mg- mg$ <---doing some housekeeping and taking out $a$ as a common factor
$a(M+m)= (M-m)g$ <---more housekeeping
$a= \frac{(M-m)g}{a(M+m)}$ <---divide both sides by $(M+m)$
Title: Re: Help me understand Mechanics!
Post by: fredrick on October 16, 2008, 10:17:37 pm: Mechanics is by far my fav spesh topic ;D wish der was more of it on the exam tho
Title: Re: Help me understand Mechanics!
Post by: xox.happy1.xox on October 18, 2008, 10:47:50 am: ^
:'(... Mechanics is ruining my life!
Title: Re: Help me understand Mechanics!
Post by: ed_saifa on October 18, 2008, 10:59:16 am: Quote from: xox.happy1.xox on October 18, 2008, 10:47:50 am
^
:'(... Mechanics is ruining my life!
It's ruining my life as well :(
Title: Re: Help me understand Mechanics!
Post by: shinny on October 18, 2008, 09:11:26 pm: It's probably the part of spesh that takes the longest to learn, but once you do, people tend to think its the easiest from what I've seen since the questions are pretty much all the same, and they can't deviate much from what VCAA standards give. i.e. Good ol' inclined plane =P It's probably also the part of spesh that's least prone to careless mistakes from my own experience too. So yeh, endure the learning part and from then on, it's pretty smooth sailing =T
Title: Re: Help me understand Mechanics!
Post by: xox.happy1.xox on October 18, 2008, 09:42:11 pm: Trying to learn Mechanics now... I can draw the diagrams now, at least... :)... But that's as far as I get :(
Title: Re: Help me understand Mechanics!
Post by: fredrick on October 18, 2008, 10:22:19 pm: If u dont know what your doing use $F_{net}=ma$

Assign one direction as positive and the opposite direction as negative(these must be parallel to each other).
If you want to find the Total force(net force) just sum all the forces together.

For example a car moving to the right with a constant force of K Newtons experiences a resistive force of K-100 Newtons, the mass of the car is 500kg and has a constant acceleration of 3ms-2. Assign right as positive and left as negative. Now the vector sum( net force) is just:
$K+(k-100)$
$2K-100$
Now using $F_{net}=ma$ we obtain
$2K-100=500(3)$
Therefore $K=800 Newtons$
Generally you apply this concept to most of the questions, except that some are on inclines where you must take the angles into consideration.
Title: Re: Help me understand Mechanics!
Post by: Mao on October 18, 2008, 10:24:04 pm: drawing diagrams is the most important step.

the next part is simply resolving into components: parallel to the direction of motion, and perpendicular to the direction of motion (which generally evaluates to zero)

when all the forces (components) are perpendicular or parallel, you find the net force in each direction (keeping in mind that some will be positive, some will be negative), and equate that to ma, where a is the net acceleration.

and now is just algebra hacking! =]
Title: Re: Help me understand Mechanics!
Post by: xox.happy1.xox on October 19, 2008, 10:21:15 pm: So basically, I just need to remember that F=ma and I can go from there? I am very slowly understanding the process. Thanks everyone for helping!
Title: Re: Help me understand Mechanics!
Post by: Pandemonium on October 20, 2008, 03:43:16 pm: Yes, however remember this point:
when a particle is undergoing constant velocity, treat the system as if it were static. this is because fnet = 0 (ie. a=0)