Solutions To Physics, Engineering, And Chemistry Essay.

Rocket Trajectory

1. Trajectory of a rocket

Consider a rocket initially located at the origin: (0,0,0). This rocket is tted with a simple transmit-ting device that relays information about the rocket’s movement.Each transmission contains a vector describing the movement of the rocket, relative to the position of the rocket at the previous transmission.

The rocket returns the following transmissions in order (a then b then c). All transmissions are in kilometres. We are not interested in what happens to the rocket after the third transmission:

a=253       b.=2113      c=2243
      3                7              10

4 10 5         430 5          4 555

(a) Suppose that at the time of the third transmission, the rocket is located at some point P. Determine the vector, OP describing the position of the rocket relative to the origin, AND determine the distance of the rocket from the origin

(b) Assume that the earth is at, and that the rocket follows a straight trajectory between transmissions. Calculate the pitch angle of the rocket (the angle the rocket makes with the xy-plane) at the time of the third transmission

(c) Calculate the de ection of the rocket’s heading on the ground between the rst and the third trans-missions (i.e., the change in direction seen on the ground)

2. The Force (required to pull a cylindrical cage up a ledge) Awakens

Princess Leia designed a large cylindrical cage for detaining Jabba the Hutt. The cage is made of two outer rings with a series of bars connecting these rings (think of a hamster wheel, but bigger). You need to roll this cage to a desirable location, but as you are doing so you come across a ledge blocking your path

To roll the cage up the ledge; you decide that the best strategy is to tie a rope to the upper-most bar

• (which is directly above the centre (O) of the cage), stand on the ledge and pull. You pull with a force F of magnitude F . The following gure outlines the scenario:

The cage has radius r and mass M. The ledge is of height h. Since the cage is quite large (taller than you), the force you exert by pulling on the rope is at an angle relative to the ground (see gure). Acceleration due to gravity is given by g.

You remember your tutor mentioning that the moment of force, or ‘torque’ produced by a force F is given by = r F, where F is the force and r is a vector from the articulation (or ‘pivot’) point (A) to the point where the force is applied.

• Determine the vector AP by calculating AO and OP

• Calculate the moment of force due to F about point A

• Calculate the moment of force due to Mg about point A

• For the cage to begin rolling up the ledge, the resultant moment (given by the sum of your solutions to (b) and (c)), must have a negative z component; indicating a clockwise turn. Show that this condition yields the following inequality
  
                                                       p

F cos( )(2r        h) > (F sin( ) + Mg)         2rh         h²

• If the radius of the cage is 3 metres, the height of the step is 1 metre, the angle is ₆ , and the magnitude of acceleration due to gravity is 9.81 m/s², what is the mass of the heaviest cage you are able to pull up the step if your pulling force has a magnitude of 500 N?

3. Chemical Catastrophe

Cage Mechanics

You work as a chemical engineer, your job is to combine various chemicals to produce pharmaceutical medicine. One day, your boss cannot make it to work, so you are in charge.

You get a panicked phone-call from a hospital; the hospital urgently needs their delivery of Medicine X. Unfortunately, you don’t have any of this particular medicine ready, and it’s one that you have not made before (normally your boss handles Medicine X ). Because of this, you are not sure how it is made (you are not sure how much of which chemicals mix together to produce Medicine X ).

You look through the documents your boss has that are relevant to this client. Although you don’t nd a recipe for Medicine X, you do nd the electronic accounting ledger where your boss has calculated the cost of producing Medicine X the last three times it was produced (most recent entry in row 1):

Cost of chemical x	Cost of chemical y	Cost of chemical z	Total cost to produce Medicine X
2	2	4	19,000
2	1	5	21,000
2	3	4	20,000

It is noted in the ledger that each chemical’s cost is given in dollars per unit of chemical. You also remember your boss ranting about how chemical prices kept changing, making it hard for him to keep pro t predictable. You also note that the hospital ordered the same quantity of Medicine X each time, which is the same quantity that they want now.

From the ledger, you can see that Medicine X is produced using chemical x, chemical y and chemical z.

a.) By forming an appropriate matrix system and applying Gaussian Elimination to an augmented matrix, show that the required quantities of chemical x, chemical y and chemical z are 2500, 1000 and 3000 respectively [2 marks].

b.)Once the boss returns, they commend you on your great work. Your boss never had the time to make more than one batch of Medicine X at a time. Your facility allows you to produce batches 5 or 10 times larger than your current batch size. They note that if you made larger batches the company could negotiate better prices from their chemical suppliers. They estimate the following discounts on the most recent ledger prices ( rst row of ledger):

25% o the price of each chemical (x, y and z) if purchasing ve times the original amount of each, 30% o the price of each chemical (x, y and z) if purchasing ten times the original amount of each.

If you make larger batches, you will need to add a preservative to the product to increase the shelf-life. The preservative costs $10,000 per unit. Based on how frequently Medicine X is ordered, the boss determines that the preservative must be added as follows:

2 units of preservative if producing ve times the original amount of Medicine X,

5 units of preservative if producing ten times the original amount of Medicine X.

i) Complete the following table describing the costs associated with producing di erent amounts of Medicine X

Batch size   Total cost of chemicals   Total cost of preservatives   Total cost of production

1   5   10

ii) Based on this information, determine the most cost e ective batch size

4. Revenge of the Cycloid

In tutorials, you studied the parametric equations for a cycloid (Vectors worksheet, problem solving exercise 3). These equations describe the path of a point on the radius of a disc which rolls without slip along a horizontal line. The parametric equations for the cycloid are:

r(t) =	y(t)	=		1	cos(t)
	x(t)			t	sin(t)

As an extension to this problem, we could consider the hypocycloid; a hypocycloid is the resulting curve traced by a point on the circumference of a circle that rolls without slip around the interior of another circle (see schematic). The position vector for a hypocycloid traced out by point P which sits on the circumference of a circle of radius r (with centre C), that rolls without slip on the circumference of the larger circle of radius R, is given by:

r(t) !

x(t)

=

(R

r) cos(t)

+ r cos

R

r

t

Rr

y(t)

(R

r) sin(t)

r sin

r t

= OP (t) =

r

a). Explain (using words and mathematics) why the hypocycloid is a horizontal segment when R = 2r

b). Find vectors describing the velocity and acceleration of the point P on the hypocycloid for general values of r and R

c). When R = 2r, show that the acceleration of the hypocycloid is proportional to the distance of P from O

a.

Distance of the rocket from origin=150

b.

c.

2.

a.

but

So,

Substituting  and  obtained in equations 1 and 2 into AB we obtain:

Optain all piont in this equations.

b.

Force

Torque by force  about point A  solve equation.

c.

Torque of w about point A is

W= mg j

From equation 2,

Then, from equation 3

d.

The resultant torque about point  equation

The equation has two z components. That is,  and

For the resultant torque to be negative, the latter component should be larger than the first component. That is,

f(2r-h)cos 0> f (sin0+ mg) 2 hr - h sqaure and root.

e.

We have:

Substituting the given variables into the inequality we get:

Dividing both sides by  we obtain

The heaviest mass that can be pulled 73.215 kg

3.

2x +2y +4z =19000

2x +y +5z= 21000

2x+3y+4z=20000

We solve the equations using Gaussian elimination as follows:

(2x +y +5z) -(2x +2y +4z) =21000-19000

-y+z=2000

Subtracting equation 1 from equation 2 we obtain

  -2y+z=2000

Then, subtracting equation 3 from equation 2 we get

2000-1000= 1000

y =1000

Subtracting equation 5 from equation 4 we obtain

Substituting  into equation 5 we get

Substituting  y and  y into equation 1 we get

Therefore

b(i)

b(ii)

batch 1 total cost 19000

batch size 10 per cost batch  19250/5 =18250

batch size 10 per cost batch 183000/10=18300

The most cost effective batch size is 5 since it has the lowest unit cost.

4.

a.

When the position of the hypocycloid will be:

 r(t) has an x-component only implying that the hypocycloid is a horizontal segment.

b.

Given that

c.

when r =r2

acce r"(t)

Hence, the acceleration is proportion to the distance oP