Downloaded from www.clastify.com by Jatin Garg 1.1 Measurements in Physics Standard/scientific form - a form of writing large numbers. Eg: 123000 = 1.23 x 105 Fundamental quantities: a unit of measurement for a measurable physical quantity from which every other unit or quantity can be derived. Derived quantities are derived from the multiplication or division of one fundamental quantity with another. Base quantity Unit Symbol Length meter m Mass kilogram kg Time seconds s Electric current amperes A Thermodynamic temperature kelvin K Amount of substance moles mol. Orders of magnitude These are numbers on a scale where each number is rounded to the nearest power of 10. For eg: 2000m = 2 x 103 ⇒ 103 (when the number is less than 5, just the power is taken) : 5000m = 5 x 103 ⇒ 10 x 10 3 = 104 (when greater than or equal to 5, estimate to 10) Orders of magnitude to memorize Diameter of proton/nucleus = 10-15 m Diameter of atom = 10-10m Diameter of the universe = 1025m Mass of an electron = 9.11 x 10-31kg ⇒ 10 x 10-31 = 10-30 Mass of the universe = 1050 Time it takes light to travel across a nucleus = s/v = 10-15/108 (from 3 x 108) = 10-23s Age of the universe = 1017 s Also know how to use metric multipliers (eg: kilo, mega, deca, deci, centi, etc.), compare ratios, values, and approximations, and estimate significant figures as appropriate. Significant figures ● A non-zero digit will always be significant (eg: 345 has 3.s.f) ● Zeros that occur between non-zero digits are always significant (eg: 303 has 3.s.f) ● Non-sandwiched zeros to the left of a non-zero digit are not significant (eg: 0.34 is 2.s.f) ● Zeros that occur to the right of a decimal point are significant (eg: 1.00 is 3.s.f.) ● When there is no decimal point, trailing zeros are not significant (eg: 400 is 1.s.f) Downloaded from www.clastify.com by Jatin Garg 1.2 Uncertainties and errors If the same thing is measured repeatedly, and you get the same value, then the result is precise. If the measured value is close to the expected value, then the result is a ccurate Types of error Random error - random fluctuations in the measured data. This is caused due to: ● Poor readability of the instrument ● The effects of changes in the surroundings ● The carelessness of the experimenter Precise experiments have small random errors. To reduce random error, take more repeats, and average. Systematic error - occur when there is something wrong with the method or the equipment. For eg: ● Zero error - When the zero setting of a piece of apparatus shifts after constant usage, ● Meter ruler edges are chipped/rounded ● Improper calibration Accurate experiments have a small systematic error. To reduce, use instruments of higher quality Absolute uncertainty in measurement Two ways of calculation (larger value is taken as the uncertainty): Option 1: (maximum measurement - minimum measurement)/2 Option 2: Take the arithmetic mean of the measurements, and the mean is considered to be the literature value. The greatest deviation of any of the measurements is taken as the uncertainty. Absolute and Fractional uncertainties Uncertainties in the form of ±Δ x are known as absolute uncertainties. These values have the same unit as the quantity and should have the same s.f. Suppose the diameter of a ball is 10cm, and there is an absolute uncertainty of 1cm. Dividing the absolute uncertainty by the value itself gives us a dimensionless value known as the fractional uncertainty. The fractional uncertainty would be written as 1/10 = 0.1 (10% as a percentage of the uncertainty). Uncertainty in equipment For an analog scale, the uncertainty is ± half of the smallest scale division. For a digital scale, the uncertainty is ± 1 in the least significant digit Downloaded from www.clastify.com by Jatin Garg Propagation of uncertainties -------------------------------------------> Significant figures in calculations ● Addition and subtraction ○ When values are added or subtracted, the number of decimal places on the least precise data determines the precision of the calculated value. ● Multiplication and division ○ Whenever you multiply or divide data, the answer should be quoted to the same number of significant figures as the least precise data Drawing graphs Error bars The table below compares the masses of plasticine balls of different diameters. Since we have just 4 data points, there is not enough distribution to use the standard deviation formula. Instead we: Even though the uncertainty is exaggerated, this calculation is accepted for smaller calculations. The uncertainties can be represented using error bars on a graph. For easier analysis, this model has to be linearized: ⍴ = mass/volume Volume = (4𝜋r3)/3 ∴ ⍴ = 3m/4𝜋r3 When rearranged: r3 = 3m/4𝜋⍴ r = d/2, so r3 = d3/8 = 3m/4𝜋⍴ d3 = 6m/𝜋⍴ 6/𝜋⍴ is a constant, so d3 must be proportional to m. Using an extension of the initial table, a linearized model can now be plotted. The length of the error bar in either direction represents the uncertainty. As per the formula, d3 is directly proportional to m. But the line does not pass through the origin. This is due to systematic error Downloaded from www.clastify.com by Jatin Garg Uncertainty in the gradient From the equation d3 = 6m/𝜋⍴, we know that the gradient of the line is 6/𝜋⍴ (y = mx+c). By calculation, the gradient of the line should be 1.797. 1.797 = 6/𝜋⍴, So ⍴ = 6/1.797𝜋 = 1.063 g/cm3 To find the uncertainty we draw the steepest and the least steep lines through the error bars: This gives the steepest grad. = 1.856 cm3/g and least steep grad. = 1.746 cm3/g. So uncertainty in the gradient using the smaller sample formula gives: (1.856 - 1.746)/2 = 0.06cm3/g If we enter the max and min values of the gradient, we obtain a max and min density: ⍴max = 6/1.746𝜋 = 1.09 and ⍴min = 6/1.856𝜋 = 1.03 Therefore the density can be written as 1.06 ± 0.03 g/cm3 Relationships (for data recording) (Think about uncertainty propagation when linearizing) Linearisation by manipulating the axis Linearisation using logs Linear Non-linear Downloaded from www.clastify.com by Jatin Garg 1.3 Vectors and Scalars A v ector is a quantity with both direction and magnitude Vector Addition Vectors can be represented using arrows. The length of the arrow is proportional to the magnitude of the vector, and the arrowhead indicates the direction of the vector. To add vectors, the arrows are placed from tip to tail, to give a r esultant. Basic trigonometry (applies to most vector problems) These equations only apply to right-angled triangles. The hypotenuse is the longest side of the triangle. The three sides of a triangle are connected by the Pythagoras theorem: Hypotenuse2 = Opposite2 + Adjacent2 For vectors in 1D, there is only forward and backward, so the magnitudes of the vectors are added to give a resultant. If +ve, the resultant direction is forward and vice versa. Subtracting vectors A negative vector is in an opposite direction to a positive vector. Hence, to subtract a vector, you just add the vector with the negative magnitude to the other vector. Scalar quantities have magnitude but no direction.
