PC1431-L02 - All figures/diagrams/illustrations used in...

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Unformatted text preview: All figures/diagrams/illustrations used in these lecture slides are copyrighted materials from the text book adopted for this module: Physics for Scientists and Engineers with Modern Physics, Serway & Beichner (5th ed.) or Serway & Jewett (6th ed.) These lecture slides are strictly for use related to the PC1431 Physics IE module. Any other use, including external circulation, is forbidden" Lecture 2 Freely Falling Objects Vectors Algebra Objectives Understand the properties of a freely falling object and able to solve related problem. Understand the properties of vectors (e.g. magnitude and direction, equal vectors, negative vector, vector addition, multiplication by scalars, etc.) To be able to represent vectors in terms of unit vectors and components for a given coordinate system, and perform vector addition. Freely Falling Objects All objects when dropped, fall toward the Earth with nearly constant acceleration (height << Re) If air resistance can be neglected, then the motion is described as free fall Free fall acceleration or acceleration due to gravity: g 9.80 ms-2 Whether an object is moving upward or downward, if it is in free fall, it will experience an acceleration downward of magnitude g. 1D motion under constant acceleration. Kinematic Equations By convention the vertical direction y is positive upward. Hence, the acceleration due to gravity (downward): y Corresponding kinematic equations: g Conceptual Example 2 A sky diver jumps out of hovering helicopter. A few seconds later, another sky diver jumps out, and they both fall along the same vertical line. Ignore air resistance, so that both sky divers fall with the same acceleration. Does the difference in their speeds remain the same? 1 Does the distance between them stay the same? Conceptual Example A tennis ball is dropped from shoulder height (about 1.5 m) and bounces three times before it is caught. Sketch graphs of its position, velocity, and acceleration as functions of time, with the +y direction defined as upward. Coordinate Systems To describe the position of a point in space 1D line requires one coordinate; 2D plane requires two coordinates; 3D space requires three coordinates A coordinate system consists of: a fixed reference point O (origin) a set of specified axes with scales and labels instructions on how to label a point relative to the origin and axes Cartesian Coordinate System (Rectangular Coordinate System) z P(x,y,z) y x Plane Polar Coordinates KIV - 3D: Polar coordinates Cylindrical coordinates Vectors Physical quantities can be scalars or vectors. A scalar quantity is specified by a single value with an appropriate unit and has no direction (temperature, mass, volume). A vector quantity has both magnitude and direction (displacement, velocity, acceleration, force etc.) Vectors quantities are specified by a number with appropriate units plus a direction. Representing Vectors y A Symbols: A (bold face, printed materials) A (hand writing) |A| = | A | = A for magnitude x Magnitude length of arrow Direction = direction of arrow Properties of Vectors Equality of vectors Adding vectors They must have the same magnitudes and point in the same direction. A+B=B+A (Commutative Law) A + (B+C) = (A+B) + C (Associative Law) Properties of Vectors Negative of a Vector A + (-A) = 0 Subtracting Vectors A B = A + (-B) A -A A A -B Multiplying a Vector by a Scalar mA is a vector that has the same direction as A and magnitude mA A 5A -B Components of a Vector Components are projections of a vector along coordinate axes. A = Ax + Ay Any vector can be described by its components Unit Vectors A dimensionless vector having a magnitude of exactly one. Particularly useful in a Cartesian (rectangular) coordinate system to introduce unit vectors i, j and k pointing in the positive x, y and z directions respectively. |i| = |j| = |k| = 1 i, j and k form a set of mutually perpendicular vectors in a right-handed coordinate system. Unit Vectors i, j and k For a 2D system, vector A can be written: Position Vector Adding of Vectors Addition of Vectors (3D) It is often easier to solve a problem by working with the vector components, and then finding the resultant vector. ...
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