VCE Specialist 3/4 Question Thread!

[SocialRhubarb]

They cancel, so what's the point of subbing in? It's easier to write k than .

[Limista]

So if I was looking for yet another k value, what should I choose to give me another different type of graph using the P equation? I suppose this question is aimed at you SocialRhubarb  I'm just asking out of curiosity.

[götze]

i been stuck on this question for days just remembered it now 

[sin0001]

Can someone explain the following simplification:
x = sin (2t + pi/2)
∴ x = cos(2t)

[09Ti08]

i been stuck on this question for days just remembered it now 

1/ find the area of that cross section:
2/ find the volume: 2680/3*80=214400/3 m^3

Can someone explain the following simplification:
x = sin (2t + pi/2)
∴ x = cos(2t)

You can think about it like this: suppose you have an angle a measured counterclockwise from the positive direction of the x axis, now it's going to have an arbitrary value cos(a), this is also the x coordinate on the unit circle. Now if you rotate it 90 degrees counterclockwise, the new angle will be a+90, and the y coordinate is sin(a+90). So why is cos(a)(x coordinate) equal to sin(a+90) (y coordinate)?  As an analogy, it is just like when you have a point (2,1) on the xy plane and you rotate it counterclockwise 90 degrees, you end up with the point (-1,2). Notice how the x coordinate of the point is equal to the y coordinate of its image.

[götze]

ily <3

[Phy124]

Can someone explain the following simplification:
x = sin (2t + pi/2)
∴ x = cos(2t)

Make sure you're familiar with the relationships between sine and cosine functions i.e.





Also that the cosine function is even and the sine function is odd i.e.





So in your example you can do it like so









WikipediA is a good learning resource for these identities

[barydos]

Good evening!
I have a kinematics-related extended response question that I need help with

Spoiler

Thanks in advance!

[SocialRhubarb]

Spoiler

Let the direction pointing vertically downwards be positive.
So we've got our first lift, and we can work out its displacement pretty easily:






Our second lift travels in the first 6 seconds.

Afterwards, its acceleration is . But, if we treat it separately from the initial first 6 seconds, notice that we can simply treat its acceleration as .

The distance that it travels when then:





At t=0, v=24, so C=24.







Difference in heights:

[e^1]

When the circle is rotated about the y-axis, a shape called a torus is obtained. If the equation of the circle is , what is the volume of the torus (without evaluating the integral)?

I'm confused on how to do this, if any could help that would be greatly appreciated!

[lzxnl]

When the circle is rotated about the y-axis, a shape called a torus is obtained. If the equation of the circle is , what is the volume of the torus (without evaluating the integral)?

I'm confused on how to do this, if any could help that would be greatly appreciated!

I love this question.
Firstly, you're rotating around the y-axis, so you'll probably want x in terms of y.
Secondly, if you look carefully, the volume of the torus is actually a volume between two solids of revolution. Draw it out. It's the volume between rotating the left half of the semicircle around the y axis and the volume between rotating the right half of the semicircle around the y axis.
So what is this volume?
(x+2)^2 = 1- (y-1)^2
x = -2 +- sqrt(1-(y-1)^2)
x^2 = 5 +- 4 sqrt(1-(y-1)^2)-(y-1)^2 i.e. two values of x^2. This is like your "volume between curves" expression
So what you need is the larger x^2 expression minus the smaller x^2 expression yielding 8 sqrt(1-(y-1)^2)
Integrate pi times this over a suitable domain and you're done. It's a semicircle of radius one multiplied by eight pi, so I'd get 4pi^2 as a result.

Or if you don't like that, note that a torus, if unravelled, becomes a cylinder. The radius is clearly the radius of the circle, i.e. 1. The length of the cylinder is slightly more complicated. In the closed form of a torus, the length of the cylinder is the circle centred at the y-axis (as that's the axis of rotation) with radius equal to the perpendicular distance from the y-axis to the middle of the circle (visualize this and it'll make sense; I'm afraid I'm hopeless at explaining these). This distance is 2 units due to the horizontal translation of the circle, so the length of the cylinder is two pi times this, or 4pi. Therefore the volume that you want is length * pi* radius squared = 4pi*pi*1^2 = 4pi^2

The length of the cylinder is the distance the centre of the circle travels when it's rotated. That might make a bit more sense.

[SocialRhubarb]

http://en.wikipedia.org/wiki/Pappus%27s_centroid_theorem

[barydos]

Spoiler

Let the direction pointing vertically downwards be positive.
So we've got our first lift, and we can work out its displacement pretty easily:






Our second lift travels in the first 6 seconds.

Afterwards, its acceleration is . But, if we treat it separately from the initial first 6 seconds, notice that we can simply treat its acceleration as .

The distance that it travels when then:





At t=0, v=24, so C=24.







Difference in heights:

Sorry, but the answer says 0
:O

Oh right i see you accidentally calculated the integral from 0 to 2, instead of 0 to 12 (despite your initial integral notation)
BUT YEAH the answer becomes zero, THANKS A LOT, you're a legend

[SocialRhubarb]

Yeah oops.

[saba.ay]

We sort of flew over these chapters in class, so I'm clueless as to what to do. Could someone please help with the attached questions?
Please and thank you.

Cross that - I understand 9, just 8 i need help with.