2. Mechanics Kinematics- the study of motion Dynamics- causes of motion Every value is a measurement and subject to uncertainty and comparable to a standard. We will use primarily the SI system of units: Length (L)  meter (m) Time (T)  second (s or sec) Mass (M)  kilogram (kg) These three fundamental dimensions are the only ones used in Mechanics.

3. Dimensional Analysis Every measured or calculated quantity has a dimension: Dimensions are not units! For example, length (L) is a dimension that can be measured in many different units. Area will always have the dimension of L 2, regardless of whether the unit of area is the ft 2, the m 2, sabins (a unit of sound absorption in) or barns (nuclear reactions). Dimensions help derive relationships in physics:

4. Intuition might tell us that the time required for an object to drop (v 0 = 0) depends upon height, the mass of the object and gravity. Dimensional analysis can help show that the mass of the object is irrelevant: t depends upon h, m, g T 1 α L A M B L C T -2C T 1 α L A M B (L/T 2 ) C L 0 M 0 T 1 = L A+C M B T -2C Solving for A, B, C will show the time depends upon the h ½ and g -½ and not mass at all!

5. An important concept in understanding the universe just after the Big Bang is the idea of Planck time, t p, which depends upon three fundamental constants: The speed of light (c), Newton’s Universal Gravitational Constant (G), and Planck’s Constant (h). (Planck’s Constant is an important value in understanding quantum physics and was actually calculated by Heisenberg). Using Dimensional Analysis, derive the value for Planck time.

6. c = 3.00 X 10 8 m/s G = 6.67 X 10 -11 m 3 /(s 2 kg) h = 6.63 X 10 -34 kg m 2 /s L = length T = time M = mass c  m/s = LT -1 G  m 3 /(s 2 kg) = M -1 L 3 T -2 h  kg m 2 /s = ML 2 T -1 t p = c i G j h k T 1 = (LT -1 ) i (M -1 L 2 T -1 ) j (ML 2 T -1 ) j

7. T = (LT -1 ) i (M -1 L 3 T -2 ) j (ML 2 T -1 ) k M 0 L 0 T 1 = (M -j +k )(L i + 3j + 2k )(T -i - 2j - k ) 0 = - j + k 0 = i + 3j + 2k 1 = -i -2j - k i = -5/2 j = 1/2 k = 1/2 t p  c -5/2 G 1/2 h 1/2 t p = Gh c 5 = 1.35 X 10 -43 s Now find a value for Planck mass (m p )

8. Planck Mass, like Planck time, depends upon the same three physical constants: c, G, h. Use dimensional analysis to determine a value for Planck mass. Ans: 5.48 X 10 -6 kg

9. Measurement and Uncertainty 41.6 cm ? 41.7 cm ? We would write: 41.65 cm or more commonly: 41.6 ±.1 cm

10. A friend borrows a valuable diamond from you to wear to the Oscars. You carefully weigh the diamond on a scale with an accuracy of ± 0.05 grams and get a reading of 8.17 g. When the diamond is returned, you weigh it again using the same scale and get a reading of 8.09 g. Is it the reasonable to assume that it is the same diamond or a different one? 8.17 g8.19 g8.12g 8.09 g8.14 g8.04g Same- but what kind of loser weighs a diamond before loaning it out

11. An average bathroom scale has an uncertainty of ± 2 pounds. A cat placed on the scale weighs 5 pounds. A person stands on the scale and weighs 225 pounds. Both of these readings were made with the same scale so they have the same precision, right? Cat: 5 ± 40% Person: 225 ± 2% Use a measuring device to best fit the quantities.

12. In calculating with imprecise values, the amount of uncertainty changes with the type of calculation: σ stands for the uncertainty (±) of a measurement z = x + y or z = x - y σ z = σ x 2 + σ y 2 If calculating the propagation of error using similar errors, the uncertainty of the answer is simply the sum of the uncertainties:

13. 2.0 ±.1 cm added to 3.0 ±.1 cm Least possible sum within the uncertainty: 1.9 cm + 2.9 cm = 4.8 cm Greatest possible sum within the uncertainty: 2.1 + 3.1 = 5.2 cm The precise answer would be written: 5.0 ±.2 cm In the lab, we will always try to measure uncertainties in order to maintain precision! In solving problems, in order to maintain precision we will follow the basic rules of Significant Figures