Chapter Readings (2,3,4)
Chapter Readings (2,3,4) PHSX 205-001
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This 9 page Bundle was uploaded by Rebeka Jones on Sunday September 18, 2016. The Bundle belongs to PHSX 205-001 at Montana State University taught by Dr. Greg Francis in Fall 2016. Since its upload, it has received 186 views.
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Date Created: 09/18/16
Chapter 2: One Dimensional Kinematics Kinematic – the description of motion, without concern for the cause of the motion One dimensional motion is motion along a straight line. In order to define where something is you can check the position of an object against the ruler and note the reading. The zero of the ruler is the point of origin. We have the right to choose which direction is positive and which direction is negative relative to the origin. This is call orientation. The combination of orientation and origin is called coordinate system. As an object changes position we say that it is undergoing translation. The amount it moves is the displacement (D). displacement involves distance and direction. In one dimension the direction is positive or negative. *delta (Δ) indicates change Δx = f inal nitial Distance – absolute value of displacement (has no sign) Speed – how fast we are going regardless of direction Velocity – how fast and what direction we are going *in one-dimension direction can be specifically positive or negative – direction is based off the coordinate system. Instantons velocity – what the speedometer reads with a sign for direction Uniform Motion – when your instantaneous velocity doesn’t change *velocity is officially m/s average velocity – average over time ∆???? ???? − ???? ????▯▯▯▯▯▯▯ = = ▯▯▯▯▯ ▯▯▯▯▯ ∆???? ▯▯▯▯▯ − ????▯▯▯▯▯▯ average velocity cannot give you the instantaneous velocity at a specific time If you want to relate the instantaneous velocity at a specific time to a distanced traveled, you must use a very short time interval *δ = very small change ???? = ???????? ▯▯▯▯▯▯▯▯▯▯▯▯▯ ???????? The standard unit for acceleration is m/s/s. The common convention is to write it as ????/???? . But this is still meters per second per second. Acceleration – change in velocity *it is wrong to assume that positive acceleration always means speeding up while negative always been slowing down. Everything is relative to the coordinate system. *in what direction is it going – velocity one way to better understand one dimensional motion is to graph the position of velocity of an object. -if equal distance in equal time it will produce a straight line -if it is going faster the slope will increase -if it is going slower the slope will decrease 2 - a flat line indicates 0 velocity Process to solve kinematics problems 1) Identify explicitly the instant in the problem that you will consider the initial time and the instant you will consider the final time. 2) Define the coordinate system you will use though out the problem . 3) Identify as many of the five variable as you can from the problem, and identify what variable you are trying to find. – the unknown For a problem to be solvable you need to know at least 3 4) Find the equation that has the variable you are trying to find and the variables that you know 5) Solve for the variable you want to find and substitute the values from the problems into the equation. ????▯= ???? +▯????∆???? ∆???? = 1 ???? + ???? ∆???? 2 ▯ ▯ 1 ∆???? = ???? ∆▯ + ????∆???? 2 ????▯= ???? +▯2????∆???? 3 Chapter 3: Vectors Principle of superposition If a body is subject to two or more separate influences, each providing a characteristic type of motion, it responds to each without modifying its response to the other. Vector quantities are those that have both magnitude and direction. They are not completely specified until both the magnitude (how far) and the direction (degrees of where) are given. –displacement, velocity, force *things that do not have a direction and things that are completely described with a signal number are called scalars Example: Temperature Vectors are denoted with a letter that has an arrow drawn over it. The same letter without the arrow will represent the magnitude of the vector. *a negative vector is just in the opposite direction. Vectors added up are called the resultant *adding vectors is not the same as adding scalars When multiplying a vector, you are simply making a vector that many times longer. The magnitude changes not direction. If we start with two vectors we can find δ???? graphically by moving one of the vectors so that it is heel to heel with the other vector and then compare the two. So, the way to think of the difference vector is that it is the vector that must be added to the initial vector to turn it into the final velocity vector. You can also add the inverse vector (it is the same length as the original vec tor just points in the opposite direction) h o Θ a The tangent of the angle Θ is defined as the ratio of the opposite side to the adjacent side. Then take the inverse tangent function. There are two ways to write a vector. We can specify a single direction and measure angle from that direction and we can write t he magnitude and angle in curly brackets. Or we can choose two mutually perpendicular directions and then write the components of the vector in these direction in parentheses A vector does not have an absolute direction angle or an absolute set of components. It only has a direction angle or components after a particular choice of the orientation of the axis have been made Chapter Formulas Formulas for right triangle are: ℎ = ???? + ???? and sin???? = cos???? = ▯ ▯ ▯ tan???? = ????ℎ???????? ???? = ℎ sin???? ???? = ℎ cos???? ???? = ???? tan???? ▯ ▯ 2 Chapter 4: Two-Dimensional Kinematics It takes 1 number to specify a position on a 1-dimensional line, 2 numbers for a 2-dimensional space, and 3 numbers for a 3-dimensional space. To create a coordinate system in 2-dimensions take a point at the origin and define two perpendicular directions to orient the system. *curly notation with magnitude and direction as the two numbers; it is polar plane system Parenthesis notation with two numbers as components; Cartesian coordinates. Anything moving along a curved path has an instantaneous velocity vector whose direction is tangent to the curve, at each point of the path, and whose magnitude is the instantaneous speed of the object along the path. Acceleration is the rate of change of velocity, not speed. -acceleration includes change in direction just because something accelerated does not mean it speeds up. Acceleration is in the same direction as the velocity. -object speeds up going in the same direction Acceleration has a forward component and a perpendicular component -velocity vector gets longer and turns in direction of perpendicular component Acceleration is perpendicular to the velocity -object turns but speed is not changed Acceleration has a backward component and perpendicular component -velocity vector gets short and turns Acceleration is in the opposite direction as the velocity -object slows down going in the same direction *small perpendicular acceleration produces small change in direction, but change in the velocity magnitude is tiny, essentially negligible. Projectile motion – the motion of objects thrown through the air *can be described in 2-dimensions. Only if there are factors such as wind does it require a 3 dimension. 2 Objects in free fall experience downward acceleration of 10 m/s (gravity) Relationship between of the angles between the acceleration and velocity vectors Speeding up -less than 90° Slowing down -more than 90° 2 Constant speed -at 90° You must divide projectile motion problems into horizontal (constant speed) and vertical (10 m/s/s down) and treat the components separately. Important -Be explicit about your choices of coordinate system and variables and be careful not to make any unjustified assumptions. -Treat the two components of motion separately, but recognize the places where the answer from the analysis of one direction may determine something about the other motion. Remember that even though something is moving at a constant speed, it accelerates if its direction changes. If an object is moving along a curved path at constant speed the acceleration much always be perpendicular to the velocity. If the path is circular, the acceleration vector must lie along a radial line, since radial lines at lines that are perpendicular to the circumference of the circle points. Because acceleration always points toward the center, the acceleration of an object undergoing uniform circular motion is called the centripetal acceleration. ▯ ???? ???? = ???? *all velocities are relative to the point of origin 3 Chapter Formulas A 2-dimensional problem may be divided into two 1-dimensional problems. Then for constant acceleration a ix the x-direction and a inythe y-direction) we have ???? = ???? + ???? ∆???? Same for y just ▯▯ ▯▯ ▯ 1 substitute x with ∆???? = ????▯▯ ???? ▯▯ ∆???? 2 y 1 ▯ ∆???? = ???? ∆▯▯+ 2 ????▯∆???? ▯ ▯ ????▯▯ = ???? ▯▯ + 2???? ▯???? ▯▯ For uniform circular motion, the acceleration toward the center is ???? = ▯ The velocity addition formula for origins, B and C is ???? ▯,▯ = ???? ▯,▯+ ???? ▯,▯ 4
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