PS113 Chapter 1 Introduction and Mathematical Concepts. Galileo Galilei, Isaac Newton, James Clerk Maxwell

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1 1. The nature of physics PS113 Chapter 1 Introduction and Mathematical Concepts Physical laws are based on experiment Galileo Galilei, Isaac Newton, James Clerk Maxwell Four forces in nature Gravity, electromagnetic, weak (radioactivity), and strong (nuclear) 2. Units We will use SI units: Le Systeme International d Unites Length (meters), Mass (kilograms), and Time (seconds) Base units and derived units 3. The role of units in problem solving feet meters, miles kilometers, hours seconds Standard prefixes: milli- 1/1000 nano- 1/1,000,000,000 mega- 1,000,000 giga- 1,000,000,000 Scientific notation: milli giga and so on. Dimensional analysis: distance (x) [L] mass (m) [M] time (t) [T] Problem 3 Vesna Vulovic survived the longest fall on record without a parachute when her plane exploded and she fell 6 miles, 551 yards. What is this distance in meters? Answer: 10,160 m 1

2 Problem 6 Consider the equation v = 1 3 zxt2. The dimensions of the variables v, x, and t are [L]/[T], [L], and [T], respectively. The numerical factor 3 is dimensionless. What must be the dimensions of the variable z such that both sides of the equation have the same dimensions? Show how you determined your answer. 4. Trigonometry Definition of sin θ, cos θ, and tan θ. tan θ = h o /h a Using inverse trigonometric functions. For example θ = tan 1 ( ) 1 2. Pythagorean theorem h 2 = h 2 o + h 2 a Problem 12 Problem 19 You are driving into St. Louis, Missouri, and in the distance you see the famous Gateway-to-the-West arch. This monument rises to a height of 192 m. You estimate your line of sight with the top of the arch to be 2.0 o above the horizontal. Approximately how far in kilometers) are you from the base of the arch? The drawing shows sodium and chlorine ions positioned at the corners of a cube that is part of the crystal structure of sodium chloride (common table salt). The edge of the cube is nm (1 nm = 1 nanometer=10 9 m) in length. Find the distance (in nanometers) between the sodium ion located at one corner of the cube and the chlorine ion located on the diagonal at the opposite corner. 5. The nature of physical quantities Scalars Vectors magnitude only magnitude and direction 2

3 6. Vector addition and subtraction A + B =? head-to-tail head-to-tail If A points purely along the x direction and B points purely along the y direction, and R = A + B, then the magnitude of R = A 2 + B 2 and the direction of R is θ = tan 1 (B/A) Watch out!! The tan 1 function is only defined between -90 o and +90 o. You have to use some common sense when reporting the final angle. Figure 1: The sum of two vectors A and B forming a right-triangle If A and B are not perpendicular to each other, then the resultant vector is still R = A + B however, R A 2 + B 2 and θ tan 1 (B/A) A B =? is the same as A + ( B) Still use the head-to-tail rule 3

4 Problem 23 Two workers are trying to move a heavy crate. One pushes on the crate with a force A, which has a magnitude of 445 newtons and is directed due west. The other pushes with a force B, which has a magnitude of 325 newtons and is directed due north. What are the magnitude and direction of the resultant force A+B applied to the crate? (b) Suppose that the second worker applies a force -B instead of B. What then are the magnitude and direction of the resultant force A-B applies to the crate? In both cases express the direction relative to due west. 7. The components of a vector A = (A x, A y ) A x = A cos θ A y = A sin θ B = (B x, B y ) B x = B cos θ B y = B sin θ Note: measure all your angles with respect to the x axis. The magnitude of A is A = A 2 x + A 2 y The direction of A is θ = tan 1 (A y /A x ) Problem 40 Your friend has slipped and fallen. To help her up, you pull with a force F, as the drawing shows. The vertical component of this force is 130 newtons, while the horizontal component is 150 newtons. Find (a) the magnitude of F and (b) the angle θ. Answers: 198 N and 40.9 o 8. Addition of vectors by means of components C = A + B C = (Cx, C y ) = (A x + B x, A y + B y ) The magnitude of C is C = Cx 2 + Cy 2 The direction of C is θ = tan 1 (C y /C x ) 4

5 Problem 47 Two forces are applied to a tree stump to pull it out of the ground. Force F A has a magnitude of 2240 newtons and points 34.0 o south of east, while force F B has a magnitude of 3160 newtons and points due south. Using the component method, find the magnitude and direction of the resultant force F A + F B that is applied to the stump. Specify the direction with respect to due east. 5