Time: 120 minutes
Number of Questions: 20

General Instructions

• Do not open this booklet until you are told to do so.
• Fill in your name, address, phone and school above now!
• You must show your multiple choice answers on the Response Sheet. USE HB PENCIL! These multiple choice answers will be computer marked to determine certificate winners. Those papers with superior computer scores will be hand marked to determine prize winners.
• You must give full written solutions on these pages. These will be hand marked to decide who wins the cash prizes and scholarships. No written solutions => no prize or scholarship!
• Don't expect to be able to answer all the questions. If you finish half, you are doing well.
• Don't guess at answers!  There are penalties for wrong answers. Random guessers should expect 0, or even a negative result.
• Aids permitted: rulers, calculators (with memory cleared), blank scratch paper.

Helpful Hints and Data

• Average value g = 9.81 m/s²
• Ideal Gas Constant R = 8.314 J/mole•K
• Use radius of earth (sphere) = 6370 km
• Use c = 3 x 10⁸ m/s
• Use Coulomb's Constant k= 9 x 109
• Atmospheric Pressure = 101.3 kPa
• All questions are of equal value, but some are tougher than others.
• Some questions contain more data than you need to solve the problem.
   

Questions

1.  Computers convert all information they process into "binary" code, a number system consisting only of the numbers 0 and 1. These 0's and 1's correspond directly to the on-off states of electric current in physical devices used to process information inside a computer. To store a binary sequence onto a magnetic disk, e.g., a floppy disk, a read-write head inside the computer is used to convert pulses of electric current into a pattern of magnetic bands on the disk. The read-write head is made of an iron core with a wire wrapped around it. When a current is sent through the wire in one direction, a corresponding direction of magnetic field is induced in the core, which magnetizes a band on the surface of the disk. Each binary digit (0 or 1), or "bit", requires two bands. The first band of each bit is always in the opposite direction of the previous band to indicate where a new bit begins. The second band is in the same direction as the first if the bit is a 0, and in the opposite direction if the bit is a 1. Adjacent bands representing the binary string are stored sequentially onto the disk as it spins past the read-write head. For example, the number 13 base ten (1x10¹ + 3x10⁰), becomes the binary number 1101 (1x2³ + 1x2² + 0x2¹ + 1x2⁰) and is stored as < > < > < < > < (where the arrows indicate the direction of the magnetic field for each band).

To store the decimal number 26, how many changes in current direction through the wire surrounding the iron core of the read-write head are required? Assume that there are no current direction changes before the first band and after the last band.

Your answer choices for question 1 are:
A: 5  B: 7   C: 4  D: 6   E: 10

Solution
 

2. A little boy retelling what he did at school one day exclaimed "I threw a ball way up into the sky at recess and it took almost forever to come down". When pressed to be specific, he reflected and said "well - at least an hour". Assuming that this little boy can throw a baseball vertically upward with an initial speed equal to 20 m/s (about half the initial speed realized by a professional baseball pitcher) and that gravity is the only force acting on the ball, how long would it take for the ball to return to its starting point? Your answer provides a back-of-the-envelope estimate for "forever" as far as young children are concerned! Of course, a rigorous calculation would include the impact of the drag force that complicates the calculation but as it turns out does not affect the result significantly.

Your answer choices for question 2 are:
A: 1 second B: 2 seconds C: 4 seconds D: 8 seconds E: 16 seconds

Solution
 

3. Icebergs floating through the Hibernia oil and gas field off the east coast of Canada can possess masses exceeding 250,000 tonnes. Icebergs pose problems for drilling and production platforms operating there because they can rupture underwater piping systems and damage the platforms themselves. The potential hazard associated with icebergs is not apparent because only a small fraction of an iceberg is above the water surface. Determine the volume fraction, above sea level, of an iceberg floating in seawater. The density of ice at 0 ^OC is 919 kg/m³ and the density of seawater is 1,030 kg/m³.

Your answer choices for question 3 are:
A: 0.017 B: 0.080 C: 0.108 D: 0.117 E: 0.121

Solution  

4. Jason Sears, the winner of the 1997/1998 da Vinci contest, suggested this question. A physics teacher balances on a unicycle at a point on the centre line of a large half-pipe, with the pedals of the unicycle at equal height. Some mean kids who are good at physics come and slowly roll the half-pipe, while the teacher continues to balance but does not pedal. They stop rolling when the teacher can no longer keep the unicycle from rolling. Find the horizontal distance between the original centre line of the half-pipe and the teacher's centre of mass when the teacher can no longer keep the unicycle form rolling.

Useful data: The coefficient of friction of the unicycle tire is very large.

Your answer choices for question 4 are:
A: (0.5 p-1) R + r meters B: a meters C: (R + r) meters D: (0.5 p-1) R +a meters E: (R + a) meters

Solution
 

5. Processes for the extraction of antibiotics from plants as well as caffeine from coffee beans exploit the fact the solubilities of solids in compressed gases first decrease then increase with pressure. This occurs because at low pressure the apparent molar volume of solids in gases, V, is larger than the molar volume of the solid, v. At high pressure the situation is reversed i.e., the apparent molar volume of the solid in the gas phase is lower than the molar volume of the solid. Expressions for this apparent molar volume, V, and solubility, S, of a solid in a gas phase as a function of pressure are given below where P is pressure and the coefficients a, b, c, d, e, f, are constants. Estimate the molar volume of the solid.
 

and  where 
 

Your answer choices for question 5 are:
A:B:C:

D: E:

Solution
 

6. Gases used in Engineering laboratories are typically stored in high pressure cylinders. These cylinders must be treated with care and securely fastened to lab benches, or walls. Most students and new laboratory workers do not appreciate the importance for doing so. Accidents do happen! When these cylinders topple the valves frequently break creating a rocket like projectile with a very high initial acceleration. If a 50 litre cylinder with a mass of 25 kg that contains 350 moles of nitrogen at 295 K is toppled and the valve is broken creating a hole with a 0.02 m diameter, what is the initial acceleration of the gas cylinder?

The gravitation acceleration constant, g, is 9.81 m/s² and the molar mass of N₂ is 0.028 kg.

Your answer choices for question 6 are:
A: 11g  B: 4g  C: 5g  D: 22g  E: 16 g

Solution

7.   Cables are frequently used to stabilize antennae - particularly in windy and remote locations. If the tensile force in cable AB is 44 KN what must the tensile force in cable AC be so that the resultant force acting on the antenna at point A is vertical?

Your answer choices for question 7 are:
A: 25 KN   B: 30 KN   C: 35 KN   D: 40 KN   E: 50 KN

Solution
 

8.  The disposal of packaging and other forms of waste is a significant concern. Without changing consumer habits engineers could redesign container shapes to minimize the amount of material employed. For example, soda pop containers are cylinders with a height to diameter ratio of 3 to 1. Soup cans have height to diameter ratios of 2 to 1. Tins for tuna and Sockeye salmon have height to diameter ratios of 1 to 2. Cylindrical containers with a height to diameter ratio of 1 to 1 use the least amount of material to contain a volume. What percentage of packaging material would be saved if soda pop were packaged in cans with a height to diameter ratio of 1 to 1 instead of the current 3 to 1?

Your answer choices for question 8 are:
A: 2 % B: 5 % C: 9 % D: 11 % E: 15 %

Solution
 

9.  An electronic flash bulb for use in photography typically uses one or more capacitors to store electrical energy for rapid discharge into a gas filled tube. The high voltage discharge turns the gas into a plasma and a brief burst of light results. Most photographic flash bulbs charge capacitors using a 300 volt DC source. In order to vary the output of the flash, capacitors are switched in and out of the circuit in banks. The switch that selects the output level of the flash bulb is called the bank switch. For simplicity, assume you have only two 800 microfarad capacitors, and a bank switch so you can connect either one or both to the flash bulb circuit. When you want the flash to fire at full power, you close the switch, and allow both capacitors to charge. You then take a picture. If you want the next picture to be exposed at half power, you open the bank switch, so that only one capacitor is charged.

The circuit design below is incomplete because if you accidentally close the bank switch when one capacitor is fully charged and the other is fully discharged i.e.; you change your mind about the flash level required before taking a picture, energy is released into the circuit, instead of the flash bulb and the circuit is destroyed. Fragments of hot molten metal from the bank switch fly outwards. Light and sound are also emitted. For the case depicted in the drawing, how much energy is inadvertently released into the circuit if the bank switch is closed?

The charge of a capacitor (coulombs) = C*V. The energy stored in a capacitor (Joules) = 0.5 CV², where C is the capacitance of the capacitor (farads) and V is voltage (volts).
 
 

Your answer choices for question 9 are:
A: 6 Joules B: 12 Joules C: 18 Joules D: 24 Joules E: 36 Joules

Solution
 

10.   A parametric plot is a plot of two functions of the same "parameter". For example, a parametric plot of a circle is a plot of r*sin(t) as a function of r*cos(t), where r is a constant and t is an arbitrary parameter. Thus points along the parametric plot (e.g. points that are on the circle) have coordinates: (r*cos(t),r*sin(t)). Comparametric plots are a special kind of parametric plot that is useful in digital signal processing. For example, comparametric plots are used in processing speech signals, music, pictures, and video signals, and have even been used for the analysis of crime scene videos in the well-known Just Desserts murder case.

A comparametric plot is a plot of a function against a dilated (stretched) or contracted (squashed) version of itself. Thus the comparametric plot of a function, f, is a plot of f(k*t) as a function of f(t), where k is the dilation or contraction (stretching or squashing) ratio. Imagine comparing a passage of music against a version of the same passage of music played twice as fast. Coordinates of points on a comparametric plot are given by ordered pairs (f(t), f(k*t)).

If f(t) = 5 + 6t, and k=2, what is the form of the ordered pairs and is the resulting comparametric plot a straight line?
 

Your answer choices for question #10 are:
A: (f, 2f-5) and yes the comparametric plot is a straight line.
B: (f, 2f-5) and no the comparametric plot not a straight line.
C: (f, 2f+5) and yes the comparametric plot is a straight line.
D: (f, 2f+5) and no the comparametric plot is not a straight line.
E: (f, 2f) and yes the comparametric plot is a straight line.

Solution
 

11.   In the March 4th edition of the Globe and Mail there was an article concerning "smart" housing where the proposed roofing material comprised photovoltaic panels that were an integral part of the building instead of being mounted on it. A British Columbia Institute of Technology researcher was quoted in the article as saying "the power generated by this grid connected solar array is approximately 2 kilowatts per hour which provides virtually all of the household power needs". Is there a problem with the quote?

Relevant facts: -Electricity is sold by the kilowatt-hour (1000 watts* 3600 seconds = 3.6 x10⁶ Joules)
-Solar power arrays do not generate electricity at night!
- Households typically consume between 600 kilowatt-hours (one person) and 1700 kilowatt-hours (six people) of electricity per month to operate electrical appliances and lighting, if air conditioning is ignored.

Your answer choices for question 11 are:
A: No, there is no problem with the quote.
B: Yes, there is a problem with the quote. The researcher is mistaken. Two kilowatts per hour does not meet typical household electrical energy needs.
C: Yes, there is a problem with the quote. The researcher was misquoted. The sentence should read "...2 kilowatt-hours per hour..." or "...2 kilowatts..." rather than "...2 kilowatts per hour..." but is otherwise correct.
D: Yes, there is a problem with the quote. The researcher was misquoted as noted in "c" but the researcher is also wrong. A significant surplus of electricity is generated except for households with large numbers of occupants.
E: Yes, there is a problem with the quote. The researcher was misquoted as noted in "c" but the researcher is also overly optimistic. The energy produced by the solar array only appears to meet or exceed the typical electrical energy requirements of households with a single occupant.

Solution
 
 

12. Fine particles of titanium dioxide, a white inorganic solid, are used as a filler in paints and fine papers. For example, without it, paint would not be opaque. In a new pigment plant that is to produce 4000 kg/hour of dry titanium dioxide product containing less than 100 ppm sodium chloride, an intermediate stream comprising a suspension of TiO2 particles, 40 wt %, dissolved sodium chloride, 20 wt %, and water, 40 wt %, must be washed with fresh water and filtered to reduce the salt content to the finished product specification. The design engineers wish to minimize water consumption and have opted to wash and filter in several stages. The vacuum filters selected remove 75 % of the water present. They propose to filter first then wash then filter, then wash etc. for a total of 5 filtration steps and 4 wash steps. Residual water in the titanium dioxide leaving the 5^th filter is then evaporated. What is the final sodium chloride concentration (in ppm) in the dry titanium dioxide and does their proposed filter + wash scheme meet the design specification?

Your answer choices for question 12 are:
A: The residual sodium chloride concentration is about 5 ppm. The design specification is met.
B: The residual sodium chloride concentration is about 50 ppm. The design specification is met.
C: The residual sodium chloride concentration is about 500 ppm. The design specification is not met.
D: The residual sodium chloride concentration is about 100 ppm. The design specification is met.
E: The residual sodium chloride concentration is about 2000 ppm. The design specification is not met.

Solution
 

13. Large office buildings require cooling year round, in part because they contain computers and other office equipment that generate heat. Toronto District Heating Corporation announced recently that it plans to take cold water from Lake Ontario and use it to cool heat exchange fluids in office buildings in the downtown core of Toronto. The warmed water would then be discharged into the municipal water intake system where once treated the water would comprise part of the potable water supply for the City of Toronto. Water near the bottom of Lake Ontario, 70 to 85 meters below the surface, remains at 4 ^oC throughout the year and is ideally suited for this purpose. The corporation plans to build a pipeline system at a cost of $120-million. The capacity of the system is such that 25% of the buildings in the downtown core can be served, resulting in a 90 % reduction in the electricity consumption for air conditioning for the customers. If this power were produced using natural gas, the proposed scheme would reduce generating-station emissions of carbon dioxide by 40,000 tonnes per year.

If electricity generation from natural gas has a thermal efficiency of 35 %, and 5 % of the electricity generated is lost during transmission, what is the value of the electricity displaced annually by the proposed project?

Data: The combustion enthalpy for methane is 800 KJ/mole where gaseous water and carbon dioxide are the only products.
The molar masses of O, C, H are 16, 12, 1 grams respectively.
The price of electricity is $ 0.06/kilowatt-hour or ($0.0167/MJ) Your answer choices for question 13 are:
A: $1-million/year B: $2-million/year C: $4-million/year D: $8-million/year E: $16-million/year
Solution
 

14.   We have all had the experience of touching hot car roofs left outdoors on sunny days. Car roofs receive energy from the sun - "solar irradiation". Only a fraction of the incident energy available is absorbed. The balance is reflected. Energy absorbed by a roof is lost to the surroundings by convection (e.g., by transfer to cool air moving past the roof), by conduction (e.g., by transfer to other objects touching the roof - your hand!) and by radiation. Use the data and equations below to estimate the temperature experienced by a cat on a hot tinny roof!

The sorbed fraction of the solar irradiation is ~ 450 W/m²

Air temperature is 300 K.

Convective heat loss (W/m²):

Radiative heat loss (W/m²):

Heat loss by conduction:
Assume that the roof is well insulated and ignore losses due to conduction!
 

Your answer choices for question 14 are:
A: 343 K B: 351 K C: 369 K D: 383 K E: 390 K

Solution
 

15.  The value of g, the acceleration produced by gravity at the surface of the earth, is not constant over the entire surface of the earth. For example, there is a smooth change in the sea level value of g from 9.832 m/s² at the poles to 9.780 m/s² at the equator. This general variation of g is due to the departure of the earth’s shape from a perfect sphere (the radius of the earth is 21 km larger at the equator than at the north pole) and the centripetal force due to the rotation of the earth reaches a maximum at the equator. If you assume that the density of the earth is constant and that the radius of the earth at the equator is 6370 km, what percentage of the decrease in the value of g, measured at sea level, from the poles to the equator can be attributed to the shape of the earth?

Your answer choices for question 15 are:
A: 25 % B: 35 % C: 45 % D: 55% E: 65%
Solution
 

16.   The calculation of forces acting on satellites orbiting planets is a complex multi-body problem. For example, the trajectory of a satellite orbiting the earth is influenced by the gravitational pull of the moon and the sun in addition to the primary earth-satellite interaction. These secondary effects must be taken into account when estimating fuel consumption for attitude correction and for other orbital trajectory adjustments over the service life of satellites. Most commercial satellites that orbit the earth occupy geo-synchronous orbits (i.e.: orbits where satellites retain a fixed position relative to the earth’s surface). Estimate the altitude of this orbit, ignoring secondary effects and then determine the maximum ratio of the gravitational force exerted by the moon on the satellite to the gravitational force exerted by the earth for a satellite occupying this orbit.

Data: The mass and radius of the earth are 6x10²⁴ kg, and 6370 km respectively.
The mass of the moon is 7x10²² kg. The distance between the centre of the earth and the centre of the moon is 3.84x10⁵ km.
The gravitational constant G has a value of 6.67x10^-11 Newtons m²/kg². Your answer choices for question 16 are:
The gravitational force ratio is:
A: 0.00018    B: 0.00032    C: 0.011   D: 0.13   E: 1.00
 

Solution
 

17. Microwave ovens are now present in most kitchens. While one can argue about the quality of the flavour of many foods prepared in them, they do cook food rapidly and in an energy efficient manner. Microwave ovens have clear advantages over conventional stoves and ovens when it comes to "steaming" foods particularly vegetables. For example, spinach, arugula, leek, chard, rapini are all cooked more rapidly, using far less energy and with better flavour retention than realized using conventional stove-top steamers. Little or no added water is needed to steam vegetables in a microwave oven because the microwaves transfer energy directly to the water within the vegetables and little energy is wasted heating the container, oven or room. To steam vegetables in a conventional stove-top steamer, a lot of water is evaporated much of which is used to heat the steamer and keep it hot, some of the water evaporated escapes to the room. Energy is lost to the stove, the pot and the room by conduction, convection and radiation. Only a small fraction of the energy expended is used to cook the vegetables. Using the data below, estimate the fraction of energy used by a microwave oven versus a stove-top steamer to cook three leeks.

Data:

To prepare the leeks in a microwave oven, seal them tightly in plastic wrap and cook on "high" - 500 watts for 2 minutes.

To prepare the same quantity of leeks in a stove-top steamer takes about eight minutes, and 140 g of water is evaporated in the process. Further, only two thirds of the energy supplied by the burner is transferred to the boiling water. The balance is lost directly to the room and the cooking pot by conduction, convection and radiation.

The enthalpy of vaporization of water is approximately 2300 kJ/Kg. Ignore the impact of the heat capacity of water on the calculation.

Solution
 

18.   This question is of the type "first" encountered by Archimedes. Initially, he found such questions baffling but later he was able to apply his findings to determine the gold content of a crown allegedly comprising pure gold. He is thus among the first recorded scientific expert witnesses! Your calculation is more pedestrian. An object, with a volume of 100 cm³ is suspended in a plastic bucket filled partly with water as shown below. The lower scale reads 1150 g while the upper scale reads 800 g. The mass of the bucket alone is 50 g. What is the volume of water in the bucket?

The density of water under these conditions is 1gram/cm³.

Your answer choices for question 18 are:
A: 300 cm³ B: 400 cm³ C: 800 cm³ D: 1000 cm³ E: 1100 cm³

Solution
 

19.   Industrial waste water quality requirements for discharge to the natural environment and even to municipal sewage treatment systems have become more stringent with time. This is particularly true for biological waste such as waste from food and beverage processing plants. Some companies with smaller plants formerly discharged their waste to municipal sewer systems but now find that they must treat their waste before doing so. They otherwise must pay a prohibitive waste surcharge. Few of these plants have space for traditional sewage treatment lagoons (shallow aerated ponds which can occupy several hectares). One alternative is to bore vertical shafts into the ground (a typical example would be a shaft with a 2 meter diameter and a depth of 150 m). The waste water is fed to the shaft and air bubbles are injected at the base of the shaft which mix and react with the waste. These open shafts, though expensive to construct, occupy little space and efficiently oxidize the waste. The waste water from the shaft reactor is filtered and then discharged to the municipal sewer system. One of the key design features of these shaft reactors is the air bubble size. If the air bubbles are too large the bacteria that digest the waste are sheared apart (i.e., killed) in their wake. Bubbles with a volume greater than 10 mm³ are to be avoided for this reason. For a 150 m deep shaft filled with waste water with a density of 1050 kg/m³. What is the largest safe bubble size that can be injected at the base of the shaft? You can assume that all of the oxygen in the bubbles reacts before exiting at the top of the shaft and that there are no gaseous products produced by the bacteria.

Data: air comprises 20 mole % oxygen, the temperature of the waste water in the shaft is 320 K, and atmospheric pressure is 100 kPa.

Your answer choices for question 19 are:
A:0.61 mm3 B: 0.76 mm3 C: 1.0 mm3 D: 7.6 mm3 E: 10 mm3

Solution
 

20. Hinged ladders are a convenient way to access attics. The bulk of the weight of such ladders is supported by the hinge, so they are readily raised and lowered. For the case illustrated, what force, T, must be exerted to just raise the ladder of length L, with a centre of mass at point G, off the floor? Express your answer in terms of the mass of the ladder, M, the gravitation acceleration constant, g, and the angle, q.

Your answer choices for question 20 are:
A: 0.5 Mg cosq/cos(0.5q)     B: 0.5 Mg/cos(0.5q)     C: Mg/cos(0.5q)     D: Mg cosq    E: 0.5 Mg cos(0.5q)

Solution