Dimensions - General Physics I - Lecture Slides, Slides of Physics

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Problem Solving Strategy

Slide 13 Fig. 1.7, p. Given: angle: distance: Find: Height=? height dist m m dist heightof building

. tan (tan 39. 0 )( 46. 0 ) 37. 3 , . tan = × = = =  α α Key idea: beam of light, building wall and distance from the building to the firefighter form a right triangle! Know: angle and one side, need to determine another side. NOTE: tangent is defined via two sides! θ = 39. 0  d = 46. 0 m Evaluate answer: 1. Makes sense (a 37 m building is Ok)

2. Units are correct.

Problem Solving Strategy

Dimensions

► Dimension denotes the physical nature of a

quantity

 dimension of some quantity, say, Q is denoted [Q]

► Dimensional analysis is a technique to check

the correctness of an equation

► Dimensions (length, mass, time, combinations)

can be treated as algebraic quantities

 add, subtract, multiply, divide  quantities added/subtracted only if have same units

► Both sides of equation must have the same

dimensions

Dimensions

► Dimensions for commonly used quantities

Length L m (SI) Area L 2 m 2 (SI) Volume L 3 m 3 (SI) Velocity (speed) L/T m/s (SI) Acceleration L/T 2 m/s 2 (SI)

 Example of dimensional analysis

distance = velocity · time L = (L/T) · T

Derived unit

► A derived unit is composed of combinations of

base units. Example: The SI unit of energy is the joule. 1 joule = 1 kg m 2 /sec 2 Derived unit Base units

2. Conversions

► When units are not consistent, you may

need to convert to appropriate ones

► Units can be treated like algebraic

quantities that can cancel each other out

1 mile = 1609 m = 1.609 km 1 ft = 0.3048 m = 30.48 cm 1m = 39.37 in = 3.281 ft 1 in = 0.0254 m = 2.54 cm

13 The density of air is 1.3 kg/m 3

. Change the units to slugs/ft 3 . 1 slug = 14.59 kg 1 m = 3.28 feet 3 3 3 3

2. 5 10 slugs/ft

3.28 feet

1 m

14.59 kg

1 slug

m

kg

−

⎟ =^ ×

Example 3. The density of air.

Prefixes

► Prefixes correspond to powers of 10

► Each prefix has a specific name/abbreviation

Power Prefix Abbrev. 10 15 peta P 10 9 giga G 10 6 mega M 10 3 kilo k 10

• centi P 10

milli m 10

micro μ 10

nano n Distance from Earth to nearest star 40 Pm Mean radius of Earth 6 Mm Length of a housefly 5 mm Size of living cells 10 μm Size of an atom 0.1 nm

4. Uncertainty in Measurements

► There is uncertainty in every measurement,

this uncertainty carries over through the

calculations

 need a technique to account for this uncertainty

► We will use rules for significant figures to

approximate the uncertainty in results of

calculations

Significant Figures

► A significant figure is one that is reliably known ► All non-zero digits are significant ► Zeros are significant when  between other non-zero digits  after the decimal point and another significant figure  can be clarified by using scientific notation 4 4 4

= ×

= ×

= ×

3 significant figures 5 significant figures 6 significant figures

Order of Magnitude

► Approximation based on a number of assumptions

 may need to modify assumptions if more precise results are needed

► Order of magnitude is the power of 10 that applies

Example: John has 3 apples, Jane has 5 apples. Their numbers of apples are “of the same order of magnitude” Question: McDonald’s sells about 250 million packages of fries every year. Placed back-to-back, how far would the fries reach? Solution: There are approximately 30 fries/package, thus: (30 fries/package)( . 10 6 packages)(3 in./fry) ~ 2 . 10 10 in ~ 5 . 10 8 m, which is greater then Earth-Moon distance ( . 10 8 m)!

Important!

► Order-of-magnitude estimates can be

helpful in determining whether the answer

you compute for a problem is reasonable.

Example: If you are asked to calculate the

weight of a car, and come up with an answer

of 10 lbs, you should re-check your calculation.

22 Be sure to label the axes with both the quantity and its unit. For example: Position (meters) Time (seconds)

23 Example: A nurse recorded the values shown in the table for a patient’s temperature. Plot a graph of temperature versus time and find (a) the patient’s temperature at noon, (b) the slope of the graph, and (c) if you would expect the graph to follow the same trend over the next 12 hours? Explain. Time Decimal time Temp (°F) 10:00 AM 10.0 100. 10:30 AM 10.5 100. 11:00 AM 11.0 100. 11:30 AM 11.5 101. 12:45 PM 12.75 102. The given data: