1981 hsc (3 Viewers)

[jttrulez]

How do you do Q8c (i) and (ii)and what's the difference between tangential and normal acceleration?

 

[annabackwards]

I don't have that paper. What's the question?

About normal VS tangential acceleration, Normal acceleration is also known as radial acceleration and it is towards the centre (centripetal force).

Tangential acceleration is along the tangent; perpendicular to the radius.

The site here should help.
In the 1st on that site, an is the tangential acceleration and the v for velocity occurs because of the tangential acceleration.

 

[Bunzhou]

Heres the link http://4unitmaths.com/hsc1981-1989.pdf So the normal acceleration is w^2r, then whats the tangential acceleration?

I don't have that paper. What's the question?

About normal VS tangential acceleration, Normal acceleration is also known as radial acceleration and it is towards the centre (centripetal force).

Tangential acceleration is along the tangent; perpendicular to the radius.

The site here should help.
In the 1st on that site, an is the tangential acceleration and the v for velocity occurs because of the tangential acceleration.

 

[Dumbledore]

well in the physics explanation, the normal acceleration is the centripetal acceleration towards the centre, it is the rate of change of velocity with respect to time with the inclusion of direction

the tangential acceleration is the rate of change of the magnitude/modulus of velocity (or speed) with respect to time

normal acceleration is a fundamental component of all circular motion whether its uniform or not

tangential is only part of non uniform circular motion as the particles actual speed as it moves around the circle speeds up or slows down

EDIT: as said above, the reason tangential acceleration speeds it up is because the acceleration acts along the tangent of the circle or the direction in which the particle is moving

 

[annabackwards]

Heres the link http://4unitmaths.com/hsc1981-1989.pdf So the normal acceleration is w^2r, then whats the tangential acceleration?

I don't think we need to know the value of the normal acceleration to be honest; i just know what it is but i haven't come across it in a paper yet but if it does come up it'll be probably be given.
Tangential acceleration = -r x d(1/2 x(d@/xt)/dt^2) but they'll ask you to derive it

Thanks for the links to the paper ^^

Here are the questions:
8. (i)(a) Given that ω is a complex root of the equation x^3 = 1, show that ω^2 is also a root of this equation.
(b) Show that 1 + ω + ω2 = 0, and 1 + ω2 + ω4 = 0.
(c) Let α, β be real numbers.Find, in its simplest form, the cubic equation whose roots are α + β,αω + βω−1, αω2 + βω−2.
(ii) Using induction, show that for each positive integer n there are unique positive integers Pn and Qn such that (1 +√2)^n = Pn + Qn√2.
Show also that (Pn)^2 − 2(Qn)^2 = (−1)n.

For 8ia - since w is a root --> w^3 = 1
now w^2 = (w^3)^2/3 = 1^2/3 = 1
Therefore w^2 is also a root to the equation.

For 8ib since w^3 = 1
w^3 - 1 = 0
(w -1) ( w^2 + w + 1) = 0
Therefore w^2 + w + 1 = 0
Since w is a root, w^2 is also a root (proven before)
So sub w = w^2 we get w^4 + w^2 + 1 = 0

Haven't figured out the rest yet :/

 

[Bunzhou]

6. A point P is moving in a circular path around a centre O. Define the angularvelocity of P with respect to O at time t. Derive expressions for the tangential and normal components of the acceleration of P at time t. Theres the question with tangential and normal.. :S

I don't think we need to know the value of the normal acceleration to be honest; i just know what it is but i haven't come across it in a paper yet but if it does come up it'll be probably be given.
Tangential acceleration = -gsintheota
Thanks for the links to the paper ^^

Here are the questions:
8. (i)(a) Given that ω is a complex root of the equation x^3 = 1, show that ω^2 is also a root of this equation.
(b) Show that 1 + ω + ω2 = 0, and 1 + ω2 + ω4 = 0.
(c) Let α, β be real numbers.Find, in its simplest form, the cubic equation whose roots are α + β,αω + βω−1, αω2 + βω−2.
(ii) Using induction, show that for each positive integer n there are unique positive integers Pn and Qn such that (1 +√2)^n = Pn + Qn√2.
Show also that (Pn)^2 − 2(Qn)^2 = (−1)n.

For 8ia - since w is a root --> w^3 = 1
now w^2 = (w^3)^2/3 = 1^2/3 = 1
Therefore w^2 is also a root to the equation.

For 8ib since w^3 = 1
w^3 - 1 = 0
(w -1) ( w^2 + w + 1) = 0
Therefore w^2 + w + 1 = 0
Since w is a root, w^2 is also a root (proven before)
So sub w = w^2 we get w^4 + w^2 + 1 = 0

Haven't figured out the rest yet :/

 

[untouchablecuz]

I don't think we need to know the value of the normal acceleration to be honest; i just know what it is but i haven't come across it in a paper yet but if it does come up it'll be probably be given.
Tangential acceleration = -r x d(1/2 x(d@/xt)/dt^2) but they'll ask you to derive it

Thanks for the links to the paper ^^

Here are the questions:
8. (i)(a) Given that ω is a complex root of the equation x^3 = 1, show that ω^2 is also a root of this equation.
(b) Show that 1 + ω + ω2 = 0, and 1 + ω2 + ω4 = 0.
(c) Let α, β be real numbers.Find, in its simplest form, the cubic equation whose roots are α + β,αω + βω−1, αω2 + βω−2.
(ii) Using induction, show that for each positive integer n there are unique positive integers Pn and Qn such that (1 +√2)^n = Pn + Qn√2.
Show also that (Pn)^2 − 2(Qn)^2 = (−1)n.

For 8ia - since w is a root --> w^3 = 1
now w^2 = (w^3)^2/3 = 1^2/3 = 1
Therefore w^2 is also a root to the equation.

For 8ib since w^3 = 1
w^3 - 1 = 0
(w -1) ( w^2 + w + 1) = 0
Therefore w^2 + w + 1 = 0
Since w is a root, w^2 is also a root (proven before)
So sub w = w^2 we get w^4 + w^2 + 1 = 0

Haven't figured out the rest yet :/

ill do the other ones now

 

[annabackwards]

6. A point P is moving in a circular path around a centre O. Define the angularvelocity of P with respect to O at time t. Derive expressions for the tangential and normal components of the acceleration of P at time t. Theres the question with tangential and normal.. :S

Ah, well let the circle have a centre 0 and a radius of R at a fixed point A on the circumference. For Tangential acceleration:
Let P be a point on the circle which is moving.
Let x = AP = r@ where @ is <P0A
dx/dt = v = rw where w = d@/dt
For acceleration, dv/dt = r dw/dt = r d^2@/dt^2

We know that normal acceleration is equal to centripetal force towards the centre.
So F = m x a(normal) = mv^2/r
So a(normal) = v^2/r = rw^2 (as v = rw as shown above)

 

[annabackwards]

ill do the other ones now

You are awesome
Still can't think of a nice way to do 8cii ><"

 

[Dumbledore]

Ah, well let the circle have a centre 0 and a radius of R at a fixed point A on the circumference. For Tangential acceleration:
Let P be a point on the circle which is moving.
Let x = AP = r@ where @ is <P0A
dx/dt = v = rw where w = d@/dt
For acceleration, dv/dt = r dw/dt = r d^2@/dt^2

We know that normal acceleration is equal to centripetal force towards the centre.
So F = m x a(normal) = mv^2/r
So a(normal) = v^2/r = rw^2 (as v = rw as shown above)

in uniform circular motion the angular velocity is constant or independant of time so dw/dt will just = 0 and the tangential acceleration is 0, which holds by the definition of tangential acceleration

 

[annabackwards]

in uniform circular motion the angular velocity is constant or independant of time so dw/dt will just = 0 and the tangential acceleration is 0, which holds by the definition of tangential acceleration

Ah i see. So for that question should i derive it as i did and then just say it equals 0 or leave it?

 

[Dumbledore]

i'd probably derive it just to be safe as the question specifically asks it

 

[untouchablecuz]

 

[annabackwards]

i'd probably derive it just to be safe as the question specifically asks it

Awesome, thanks

untouchablecuz, you are going to rip tomorrow's exam!

 

[Bunzhou]

Cheers guys. Shouldn't the mathetmatical induction specific an initial condition of Pn = Qn = 1 though.. because you just assumed its 1?

 

[untouchablecuz]

Cheers guys. Shouldn't the mathetmatical induction specific an initial condition of Pn = Qn = 1 though.. because you just assumed its 1?

(ii) Using induction, show that for each positive integer n there are unique positive integers Pn and Qn such that (1 +√2)^n = Pn + Qn√2.
i.e. n>=1
so P_{1} and Q_{1} are the initial conditions which we deduce from (1+sqrt(2)^1=P_1+sqrt(2)P_1

 

[Bunzhou]

How'd you get from second last step to the last step?

ill do the other ones now

 

[jet]

Because, we require that P_k+1 and Q_k+1 be integers. Now, since P_k and Q_k are integers, it follows that P_k + 2Q_k and P_k + Q_k are also integers. We're just saying that P_k+1 = P_k + 2Q_k and Q_k+1 = P_k + Q_k.

Think of it as simplifying the expression.

 

[Bunzhou]

Oh I see, makes sense.. thanks heaps

Because, we require that P_k+1 and Q_k+1 be integers. Now, since P_k and Q_k are integers, it follows that P_k + 2Q_k and P_k + Q_k are also integers. We're just saying that P_k+1 = P_k + 2Q_k and Q_k+1 = P_k + Q_k.

Think of it as simplifying the expression.

 

[Bunzhou]

Anyone manage to do part ( i )? Question is

(c) Let α, β be real numbers.Find, in its simplest form, the cubic equation whoseroots are α + β,αω + βω^−1, αω^2 + βω^−2.
[From prev parts] 8. (i)(a) Given that ω is a complex root of the equation x^3 = 1, show that ω^2 is also a root of this equation.(b) Show that 1 + ω + ω2 = 0, and 1 + ω2 + ω4 = 0.

How do you do Q8c (i) and (ii)and what's the difference between tangential and normal acceleration?