Physics 103 – Introduction to Physics I

1. Physics 103 – Introduction to Physics I Motion Forces Energy

2. First Dimensions Units Precision Coordinate Systems Vectors Kinematics Motion Variables Constant Acceleration

3. Dimensions The dimension of a physical quantity specifies what sort of quantity it is—space, time, energy, etc. We find that the dimensions of all physical quantities can be expressed as combinations of a few fundamental dimensions: length [L], mass [M], time [T], and either electric charge [Q] or electrical current [A]. For example, energy - . The physical quantity speed has dimensions of .

4. dimension SI cgs Customary [L] meter (m) centimeter (cm) foot (ft) [T] second (s or sec) second second [M] kilogram (kg) gram (g) slug or pound-mass Units International System (SI) The units of the fundamental dimensions in the SI are The SI units will be introduced as we go along.

5. Unit Conversions We might measure the length of an (American) football field with a meter stick and a yard stick. We’d get two different numerical values, but obviously there is one field with one length. We’d say that . In other words, Suppose we wish to convert 2 miles into meters. [2 miles = 3520 yards.] The units cancel or multiply just like common numerical factors. Since we want to cancel the yards in the numerator, the conversion factor is written with the yards in its denominator. Since each conversion factor equals 1, the physical measurement is unchanged, though the numerical value is changed. Note: the units are a part of the measurement as important as the number. They must always be kept together.

6. Precision & Significant Digits Instruments cannot perform measurements to arbitrary precision. A meter stick commonly has markings 1 millimeter (mm) apart, so distances shorter than that cannot be measured accurately with a meter stick. We report only significant digits—those whose values we feel sure are accurately measured. There are two basic rules: (i) the last significant digit is the first uncertain digit and (ii) when combining numbers, the result has no more significant digits than the least precise of the original numbers. A third rule is, the exercises and problems in the textbook assume there are 3 significant digits. Therefore, we never include more than 3 significant digits in our numerical results, no matter that the calculator displays 8 or 10 or more.

7. The uncertainty in a numerical value may be expressed in terms of a tolerance, as Alternatively, the uncertainty can be shown in scientific notation simply by the number of digits displayed in the mantissa. 2 digits, the 5 is uncertain. 3 digits, the 0 is uncertain. [Notice the ambiguity. Do we speak of the number of significant digits, or of the relative “place” of the uncertain digit? That is, should it be 18 or 17.6?]

8. Coordinate Systems We measure locations in space relative to a coordinate system. Firstly we select the origin of coordinates, and then the directions of orthogonal axes. Since the directions shown by orthogonal axes are mutually perpendicular, components along different axes are independent of each other. The commonly used two-dimensional coordinate systems are the Cartesian and the plane polar systems.

9. The three dimensional Cartesian coordinate system is comprised of three mutually perpendicular, straight axes, commonly denoted x, y, & z or . [We’ll talk about those hat-things later.] The spherical polar coordinate system is comprised of a radius and two angles, as shown in the figure. Notice how the polar coordinates are defined in terms of the Cartesian system. Any point in space can be uniquely specified by listing three numerical coordinates.

10. Vectors As used in Physics, a scalar is a quantity that has only one property—a magnitude. Energy, speed, temperature, and mass are scalar quantities. A vector is a quantity that has two properties—a magnitude and a direction. Displacement, velocity, acceleration, and force are vector quantities. In text or equations, vectors are denoted with either a line or an arrow on top, thusly: In diagrams, a vector is represented by an arrow. In text books, vectors are often denoted by bold-faced letters: A .Weirdly, University Physics uses both bold-face & an arrow! A is not the same as A! . . . .

11. The directions defined by the Cartesian coordinate axes are symbolized by unit vectors, . A set of unit vectors that define a coordinate system are called a basis set. Two dimensional: Components A unitvector is a vector of magnitude 1. E.g., , where is the magnitude of the vector . Often, the magnitude of a vector is indicated by the letter without the arrow on top: .

12. An arbitrary vector can be written as a sum of the basis set unit vectors. Direction cosines emweb.unl.edu

13. Adding vectors The sum of two vectors is also a vector. Drawn to scale. A vector may be multiplied by a scalar. This affects the magnitude of the vector, but does not affect its direction. The exception to this rule is multiplication by –1. That leaves the magnitude unchanged, but reverses the direction.

14. Vector Products scalar (or dot) product—result is a scalar; the operation is symbolized by a dot. The angle is the angle from to . Note: and . Vector (or cross) product—result is a another vector; the operation is symbolized by a cross, . , direction perpendicular to both and according to the right-hand-rule. [Use the three-finger version.]

15. Kinematics Simply describe the motion of an object.

16. Motion Variables The displacement vector, , points from the origin to the present location of the particle. If a particle is at at time and at at some later time , then we say the change in displacement is . Likewise, the elapsed time ortime interval is . The average velocity during the time interval is defined to be . It’s the time-rate-of-change in the displacement. In terms of vector components, we’d write . The instantaneous velocity is defined to be . Similarly, the average acceleration is .  The instantaneous acceleration is .

17. Constant Acceleration Commonly, We have four equations that each relate three of the motion variables.

18. Space-time Mathematically we can treat time and space on the same footing. The displacement vector in space-time has 4 components. The scaling factor c is needed to make the units of all 4 components the same, e.g., meters. The geometry of space-time is not Euclidian, but is non-Euclidian. Therefore,

19. time, t (seconds) acceleration, ax (m/s2) 0 - 10 2 10 – 40 0 40 - ? -4 Example: a train traveling on a straight and level track starting from rest; ends at rest. What is the total displacement?Segment 1: We are given the acceleration, elapsed time and initial velocity — vxo = 0 m/s. Segment 2: To find the total displacement at the end of the second segment, we need the velocity component at the end of the first segment. Segment 3: For this segment, we know x2, vx2, vx3, and ax, but not .

20. Example: A hot air balloon is rising at a constant speed of 5 m/s. At time zero, the balloon is at a height of 20 m above the ground and the passenger in the balloon drops a sandbag, which fallsfreely straight downward. We observe that . What are the height of the sandbag and its velocity as functions of time? Free Fall! What is the y-component of the sandbag’s velocity when it hits the ground?

21. How long does that take? Alternative solution for the elapsed time:

22. Projectile Motion Constant acceleration, in two dimensions.

23. Vector equations: Component equations: Notice:The y-component of is ay = -g.

24. Example: How long does it take to reach maximum height, ymax?At maximum height, vy= 0 m/s What is the maximum height?

25. When is the projectile at y = 25m?

26. time (s) velocity (m/s) 0.910 5.61 What are the velocity components then, at t = 0.910 s and t = 5.61 s?

27. Example: How far does the object travel in the x-direction? We need to know the elapsed time, t. The total elapsed time is the time it takes to go up plus the time it takes to come down.Previously, we found that the time to reach maximum height was t = 3.26 s. The total time, then, is 2x3.26s = 6.52 s. [Verify with .]

28. Example: What are the velocity & position components at t = 3 seconds?

30. Uniform Circular Motion Curvilinear motion – not in a straight line. Envision an object having, at the moment, a velocity , and subject to an acceleration . . We might decompose the acceleration into components parallel to and perpendicular to the velocity vector. The parallel acceleration component affects the speed of the object, while the perpendicular component affects the direction of the velocity vector, but does not change its magnitude. At any instant, the velocity to tangent to the curve.

31. Circular motion Uniform circular motion refers to motion on a circular path at constant speed. While the magnitude of the velocity is constant, the velocity vector is not constant. The same is true of the acceleration vector—its magnitude is constant but its direction is not. However, the acceleration is always directed toward the center of the circular path. The component of acceleration parallel to the velocity vector is zero. The acceleration component directed toward the center of the circle is called the centripetal acceleration.

32. Let the origin be at the center of the circle, as shown.

33. Consider two successive displacement and velocity vectors. By the definition of uniform circular motion, In the limit as , & . . Both are isosceles triangles, with the same angle. The centripetal or radial acceleration is always on a circular arc of radius r.

34. Second Dynamics Newton’s “Laws” Energy Momentum Conservation

35. Dynamics Relationships among Motion and Force and Energy.

36. Newton’s “Laws” of Motion “An object in uniform motion remains in uniform motion unless it is acted upon by an external force.” [In this context, uniform motion means moving with constant velocity.] “The change in motion of an object is directly proportional to the net external force.” . “For every action, there is an equal and opposite reaction.”

38. A force is an external influence that changes the motion of an object, or of a system of objects. We find that there are four fundamental forces in nature, gravity, electromagnetic force, and the strong and weak nuclear forces. All particles of matter interact through one or more of these four fundamental forces. All other types of forces that we might give a name to are some manifestation of one of the fundamental forces. Dimensions of force are The SI unit of force is the Newton (N).1 N = kg m/s2

39. Fundamental concepts: • Space and time • Matter and energy Macroscopic objects—collections of many atoms & molecules. Molecules—combinations of several atoms; chemical substance. Atoms—combinations of protons, neutrons & electrons; chemical element. Subatomic particles—protons, neutrons, electrons, et al. A particle is an idealized object that has no shape or internal structure. Any object may be treated as if it were a particle depending on the context.

40. Two of the attributes of matter are • resists changes in its motion—matter has inertia, and • ii) a force acts between any two pieces of matter—material objects or particles • exert forces on each other. The quantitative measure of inertia is called the inertial mass of a particle. Imagine two particles exerting equal and opposite forces on each other. We observe their accelerations. We write Newton’s 2nd “Law” in mathematical form:

41. Classical assumptions: i) time is independent of space and is absolute. ii) 3-d space is Euclidian—”flat.” Unless a particle is moving at a very great speed, its momentum is approximately Further, if the particle’s mass is unchanging, then

43. Equilibrium Should the vector sum of all forces acting on an object be equal to zero, then and the object is said to be in Static equilibrium Dynamic equilibrium

44. Isolated body diagram(s) An isolated body diagram is a sketch of the object only, with arrows indicating each force acting only on that object.

45. Action & Reaction Force is an interaction between two material objects. E.g., there is a gravitational interaction between the Earth and the Moon. They exert forces on each other of equal magnitudes but opposite directions.

46. Newton’s Universal “Law” of Gravitation Any two objects exert gravitational forces on each other, equal in magnitude and opposite in direction. Take care with the directions. The unit vector points from M1 to M2.The gravitational force on M2 is in the direction, toward M1.

47. Gravitational Field Let’s say M1 is at the origin of coordinates. The presence of M1 gives rise to a gravitational field that extends outward into space. An object of mass M2 located at experiences a gravitational force. In the context of the 2nd “Law” The acceleration due to gravity is

48. Near the Earth’s surface, Near the surface of another body, such as the Moon or Mars, the acceleration due to gravity is different, not 9.8 m/s2.

49. Weight Weight is the term we use to refer to the force of gravity near the Earth’s surface, or near a planetary body’s surface, or near a moon’s surface, etc. We do not measure weight of an object directly. Instead, we place the object on a scale. The number we read off of the scale is actually the contact force exerted upward by the scale on the object. If the object is in equilibrium, then we infer that the weight has the same magnitude.

50. Suppose the object is not in equilibrium. Suppose A = -g. Then N = 0. The object is in free fall, but notweightless. The term weightless is a misnomer.