brightsky's Maths Thread

[brightsky]

A lift is pulled from the basement of the building to the top floor by a cable. The lift first moves with constant acceleration, then constant velocity and finally constant retardation. If T is the magnitude of the tension in the cable during the motion and t is time, what would the graph T versus t look like?

How best to approach questions like this one?

[Planck's constant]

A lift is pulled from the basement of the building to the top floor by a cable. The lift first moves with constant acceleration, then constant velocity and finally constant retardation. If T is the magnitude of the tension in the cable during the motion and t is time, what would the graph T versus t look like?

How best to approach questions like this one?

The best approach would be to define T as a hybrid function for the three stages (using Newton's 2nd Law)
Assuming UP positive and a,b>0 :

Stage 1 : T-mg = ma   => T = m(a + g),  t0<t<t1
Stage 2 : T-mg = 0 => T = mg,                t1<t<t2
Stage 3 : T-mg = -mb => T = m(g-b),      t2<t<t3

.... and the graph should look like a descending staircase

[brightsky]

cheers! i have another one:

in general, if you're given the position vector r(t) of a particle, how would you find the angle that the path of the particle makes with a certain plane at a time when it touches that plane?

[TrueTears]

you should dot product r'(t) with some vector that is on the plane

[brightsky]

thanks! and the vector can be any vector right? so long as it is on the plane?

[Planck's constant]

you should dot product r'(t) with some vector that is on the plane

TT, I was thinking dot product of r'(t) with the normal to the plane, subtract from 90 degrees.

Still leaves a few questions though, ie how is the plane defined? How do you find t at the time of intersection? But most importantly how can you solve this problem using VCE methods

[pi]

Just out of my own curiosity (sorry for the mini-hijack brightsky), could the vector on the plane be r(t)?

[TrueTears]

you should dot product r'(t) with some vector that is on the plane

TT, I was thinking dot product of r'(t) with the normal to the plane, subtract from 90 degrees.

Still leaves a few questions though, ie how is the plane defined? How do you find t at the time of intersection? But most importantly how can you solve this problem using VCE methods

yeah same here, it depends on how the angle is defined tbh, if you dot it with a vector on the plane then it would calculate the angle relative to that vector parallel to the plane (i guess you can call it relative to the "horizontal"), if you dot it with the normal of the plane you calculate the angle made with the vertical, so it kinda depends on the question

[Planck's constant]

 Brightsky, I truly believe that someone like you would benefit enormously from UMEP Mathematics. It's not for the faint hearted, and I sort of understand what you are trying to do with your VCE, but I can't help thinking that UMEP Maths was made for the likes of you.

[brightsky]

thanks!

A plane is forced to make an emergency landing. After landing, the passengers are instructed to exit using an emergency slide, of length 6 metres, which is inclined at an angle of 60 degrees to the vertical. Passenger Jay has mass 75 kg. The coefficient of friction between Jay and the slide is 1/5. The plane came to rest so that the end of the slide is 2 metres vertically above level ground.

a) How far horizontally from the end of the slide does Jay land?

b) Find the magnitude of Jay's momentum in kg m/s, when he hits the ground.

[Planck's constant]

thanks!

A plane is forced to make an emergency landing. After landing, the passengers are instructed to exit using an emergency slide, of length 6 metres, which is inclined at an angle of 60 degrees to the vertical. Passenger Jay has mass 75 kg. The coefficient of friction between Jay and the slide is 1/5. The plane came to rest so that the end of the slide is 2 metres vertically above level ground.

a) How far horizontally from the end of the slide does Jay land?

b) Find the magnitude of Jay's momentum in kg m/s, when he hits the ground.

I will describe the method.

* Your are firstly concerned with Jay's motion on the slide. This stage of the problem is your typical inclined plane problem with a coefficient of friction.
* resolve your forces (weight, normal reaction and friction) along the slide (x direction) and perpendicular to the plane (y-direction)
* apply Newtons 1st Law in the y-direction to find the Normal reaction force, and hence calculate the Friction force
* apply Newtons second law in the x-direction to calculate the acceleration
* you now have a kinematics problem for the slide. Constant acceleration (from previous step), initial velocity zero, distance travelled 6m. Use the appropriate formulas to determine the final velocity at the bottom of the slide.

* you now have a projectile motion problem, as Jay takes-off from the bottom of the slide, at a known angle and speed (as per previous step)
* resolve the take-off velocity into its horizontal and vertical components.
* use the vertical component component of the velocity, the known constant acceleration, g, and the known distance travelled (vertical drop) to determine 1) the time taken for the vertical drop and 2) the final velocity in the vertical direction. Both of these quantities will be required later.
* use the time for the vertical drop, determined in the previous step, and multiply this by the (constant) horizontal component of the take-off velocity. This is the answer to question a)
* for question b) you use geometry (Pythagoras) to determine the speed with which Jay hits the ground. You already know the horizontal speed, it never changed since take-off, and the vertical speed when he hit the ground was found earlier. Find the resultant speed and multiply by the mass to determine the momentum.

[brightsky]

thanks argonaut!

[brightsky]

For questions which star a conveyer built moving an object up an inclined plane, can someone explain to me how friction is the force that's moving the object upwards? Or do we say so simply for convenience?

[b^3]

Think of it this way, for a normal inclined plane, with no conveyor belt, the ground is stationary, and the object moves down the slope then friction is in the opposite direction to the direction of motion. But if we have an object on a conveyor belt on the inclined plane, the difference is that the 'ground' is not stationary, and we need some force for the box to move with the conveyor belt, and this force will be the friction force between the box and the conveyor belt, because if the friction weren't there, then the box would slip on the conveyor belt and not move up the plane.

I think I kinda butchered that explanation a bit... but hope it helps anyway.

EDIT: Post number 2300

[brightsky]

Question 1

Sketch, on an Argand diagram, U = T intersection {z:z<Arg(v)} where T = {z: mod(z - 2) =< 2, z E C} and v = 3 + sqrt(3) i.

I'm confused about the notation used in this particular question. z < Arg(v) implies z < pi/6, which doesn't seem to make sense to me. Any help will be much appreciated.

Question 2

How exactly is a 'disc' defined? A question I've come across of late reads:

Sketch the disc whose boundary includes the circle that passes through two points 'blah' and 'blah', where the line joining 'blah' and 'blah' is a diameter of the disc. I conceive of a disc as a three-dimensional object, but obviously this is an incorrect interpretation in the context of this question.

Thanks!