PHYS 204-04 Mechanics Past Paper PDF January 2025

Note_01 Chap. 1 Units of measurement PHYS 204-04 Mechanics January 2025 Introduction  Physics is a study of CAUSE & EFFECT quantitatively.  Physics does not explain WHY an event happens (e.g. why dropping an object falls to the ground). It does show HOW...

Note_01 Chap. 1 Units of measurement PHYS 204-04 Mechanics January 2025 Introduction  Physics is a study of CAUSE & EFFECT quantitatively.  Physics does not explain WHY an event happens (e.g. why dropping an object falls to the ground). It does show HOW the object should fall, governed by some simple laws. There is the distinction between why and how. Facts (correlation) are not necessary the same as principle (causation) of an event. When a doctor prescribes medication for your illness, he or she does not necessarily know why the illness occurs. However, the doctor knows how certain medication will work. We still have no clue why cows can turn grass into milk with whatever enzyme.  In Physics we attempt to find out the relationship between causes an effect. Physics does not explains why (and sometime we use the word how) a physical phenomena occurs.  Number counts. Quantifying an event is important. For example, the inner surface of the tire used in a Tesla automobile is lined with a 5-mm foam. The function is to reduce the road noise. The question is by how much. Some company promotes a tire designed for EV with lower and their road tests have shown an improvement of 5~8% better gas mileage. However, they are hiding some fact from the customers. They make the thread with lower rolling friction. The law of physics also shows better mileage with worn out (bald) tires as they do have lower rolling friction. You save a few dollars at the pump, but losing control of the car in a wet pavement will cost a lot more than the savings. The advertisement is an half truth and a trade off at best. It is important to distinguish truth from half truth. There is no half truth in physics.  Most physical events involve more than a single physical law. We have to start with the basic and that is mechanics.  All physical laws are empirical and nothing theoretical. We draw conclusions from many observations and prove them in the lab with a 100% repeatable result. Only then it becomes a law.  Scientific Method implies repeatability. The Big Bang is not repeatable. It is only a theory.  In order to communicate between investigators, some standard of units are necessary.  Any physical event must be able to describe in terms of mass, length and, time, and nothing else. This course assumes you have acquired the skill of some basic high school level algebra, geometry and trigonometry. If in doubt brush it up. You will definitely need them to follow the course. Most of the basic math are given in the Appendices of your text book. Page 1 of 5 Note_01 Chap. 1 Units of measurement PHYS 204-04 Mechanics January 2025 Chapter 1. Mostly common sense. All physical phenomena are described in terms of 3 basic units of measurement. MKS units (Meter-Kilogram-Second) are practically the same as SI units. The minor difference is found in energy unit in heat (Joule vs Newton-meter).  Length. How do you know if measurements made by your ruler are accurate. Any ruler, measuring tape, must be able to trace back to some universal or secondary standard. The 1 universal definition of 1 meter is the distance travel by light in a vacuum in of 1 299 792 458 second in time. This begs the question what is the time standard of 1 second. Thus the second must be defined before the meter. For practical purpose, there is a platinum-Iridium bar kept in NRC lab (National Research Council) in Ottawa. The two scribed lines on the bar indicate exactly 1 meter. In the lab, one can calibrate length by comparing it with the wavelength of the emission by some excited gas (e.g. krypton) using a Fabry-Perrot interferometer. The wavelength of the emitted light is absolutely constant, unaffected by temperature and other environmental factors.  Second. The hyperfine transition of the ground state of Cesium 133 atoms has a frequency of 9 192 631 770 per second (>9 GHz). Very high speed electronic counters can count the pulses without losing a beat. That is the heart of the atomic clock. An error of 1 count correspond to 1 second in 300 years. The atomic clock converts counts into time.  Mass. The standard 1 kg mass is based on a piece of platinum-iridium metal stored in a vault in Sevres, France. A copy is found in NRC lab in Ottawa. Since 2019, the new definition of the kg is based on the Planck constant, velocity of light and the frequency of the transition of Cs133. The following examples show when and how to apply the SI units in problem solving. Page 2 of 5 Note_01 Chap. 1 Units of measurement PHYS 204-04 Mechanics January 2025 Problem 1-2, p.17 of textbook. A proton, which is the nucleus of a hydrogen atom, can be modeled as a sphere with a diameter of 2.4 fm and a mass of 1.67  1027 kg. (a) Determine the density of the proton. (b) State how your answer to part (a) compares with the density of osmium, given in Table 14.1 in Chapter 14. Answer This is only a model. There is no proof that the proton behaves like a rain drop. (a) 1 femto m  1  1015 m. Table 14.1 p.361 shows density of osmium is 22.6  103 kg/m3. mass, m of a 4 4 d 3  d 3  , V   r3   where d  2r is the diameter. volume,V sphere 3 3 8 6 m 6 (1.67 10 27 ) 6 1.67 1027  proton     0.23 1018 kg/m3  steel  8  103 kg/m3  d / 6  (2.4 10 )  (2.4) 10 3 15 3 3 45  proton 0.23 1018 (b)   1.0  1013. Osmium is already 3x more dense than steel. osmium 22.6 103 Note that if the quantities are expressed in SI units, it is not necessary to show the unit of each individual item. The end result will always be in the proper SI units SI units applies only to physical events. Money is not a physical event and has its own units.  Some common non-SI unit we see everyday. 1 inch  almost exactly 25.4 mm. 1 mile = 1609m  (8 / 5) km. 1 lb  (1/ 2.2) kg. Problem 1-11, p.17 of textbook. A subset of SI units. A piece of lead has a mass of 23.94 g and a volume of 2.1 cm 3. From these data, calculate the density of lead in SI unit. (kilograms per cubic meter). Answer m Density is defined as mass per unit volume,  . V 23.94 g  1 kg   100 cm  3 23.94 g  1 kg   1 000 000 cm 3           11.4  10 kg m 3 3 2.10 cm3  1 000 g   1 m  2.10 cm 3  1 000 g   1 m 3  All these dimensions are unnecessary if we just stick with MKS units. g → kg, shift left 3 places. cm3 → m3, multiply by (0.01)3. 0.02394  6  11.4  103 kg m3 2.10  10 Page 3 of 5 Note_01 Chap. 1 Units of measurement PHYS 204-04 Mechanics January 2025  Order of magnitude. Everything in life is relative. It is good to know some of these numbers. Average atomic spacing  5 1010 m  0.5 109 m  0.5 nm. Distance from the sun to earth  150  106 m. A commercial airline travel with an average speed about 950 km/hour ( 264 m/s). Speed of sound is most often quoted as 343 m/s.  All measurements, no matter how accurate, are approximations. Problem 1-36, p19 of textbook. In physics, it is important to use mathematical approximations. (a) Demonstrate that for small angles  tan   sin     where  is in radians and   is in degrees. (b) Use a calculator to find the 180 largest angle for which tan  may be approximated by  wth an error less than 10%.  (degrees) radian tan   sin   difference between  and tan  15.0 0.262 0.268 0.259 2.30% 20.0 0.349 0.364.342 4.09% 30.0 0.524 0.577 0.500 9.32% 33.0 0.576 0.649 0.545 11.3% 31.0 0.541 0.601 0.515 9.95% 31.1 0.543 0.603 0.516 10.02% We see that  in radians, tan() and sin() start out together from zero and diverge only slightly in tan    value for small angles. Thus 31.0 is the largest angle for which  0.1. tan  Sine does a bit better. At 31, the difference is about 5%. Page 4 of 5 Note_01 Chap. 1 Units of measurement PHYS 204-04 Mechanics January 2025  Dimension analysis is to express every quantity with its unit in a calculation to ensure the final result is of the correct unit or dimension. It is useful when converting non-SI units and is not a big deal! Example 1.4, p.12 of textbook. Is He Speeding? On an interstate highway in a rural region of Wyoming, a car is travelling at a speed of 38 m/s. Is the driver exceeding the speed limit of 75.0 mi/h? (mph) Answer Convert meters to miles and seconds to hour: m  1 mi   60 s   60 min  38.0      85.0 mi/h s  1609 m   1 min   1 h   conversion factors The driver is indeed exceeding the speed limit and should slow down. WHAT IF? What if the driver were from outside the United States and is familiar with speeds measured in kilometer per hours? What is the speed of the car in km/h? Answer We can convert our final answer to the appropriate units: mi  1.609 m  km 85.0    137 h  1 mi  h  conversion factor Dimension analysis is necessary only when working with non-SI units. If all quantities are in SI units, it is unnecessary to indicate the units in the expression and the result is always correct in dimensions. Another example. Given a mass = 100g and a speed of 600 m/s, find its momentum. Answer (The givens are in mixed units, but is metric). By definition, linear momentum is defined as the product of mass times its velocity 1 kg m p  mv  0.1 600  60 kg-m/s No need to write p  100 g   600  60 kg-m/s 1000 g s  Accuracy  precision. A digital voltmeter shows a reading of 4 significant numbers. The last digit represents one part per 9,999. That is high precision. However, the indicated value but could be off by as much as 10%. It comes down to trust. Instrument made by trusted brands are usually calibrated with some sort of secondary standard. Page 5 of 5