DOCSLIB.ORG

Linear Motion Review

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Motion Review Linear Motion Review Level : AP Physics Teacher : Kim ■ Displacement vs Distance 1-Dimension(=linear) A jogger starts at position A and runs in a straight line, stops at 10m, reverse direction to position B. A B 0m 6m 10m - The (total) distance the jogger traveled is 10m+4m=14meters - The displacement(=change in position) of the jogger is 6meters. 2-Dimension ex) A jogger starts at position A and runs to position B. The dotted line is the actual path of the jogger 5m A● 4m 5m ●B - The (total) distance the jogger traveled is 5m + 4m + 5m = 14meters - The displacement(=change in position) of the jogger is 10.8meters ( √102 + 42=10.8meters 풕풐풕풂풍 풅풊풔풕풂풏풄풆 ■ Average Speed (v) for linear motion is defined as v = 풕풐풕풂풍 풕풊풎풆 푡표푡푎푙 푑푖푠푡푎푛푐푒 ex) If the jogger took 10 seconds to run from A to B, then the average speed is v = = 푡표푡푎푙 푡푖푚푒 14 =1.4m/s 10 휟퐱 풙−풙풐 ■ Average Velocity(v) for linear motion is defined as v = = 휟풕 풕−풕풐 where v is velocity, Δx is displacement and Δt is time interval 휟퐱 10.8 Ex) If the jogger took 10 seconds to run from A to B, then the average velocity is v = = =1.08m/s 휟풕 10 휟풙 풅풙 ■ Instantaneous Velocity or Speed vx ≡ 풍풊풎 = ( linear) 휟풕→ퟎ 휟풕 풅풕 *~For linear motion, instantaneous velocity and instantaneous speed is similar. Instantaneous speed is just how fast an object is moving regardless of the direction, while instantaneous velocity represents how fast and direction. ■ Acceleration 휟풗 풗−풗풐 훥푣 풅풗 Average acceleration is defined at a = = and instantaneous acceleration is ax ≡ 푙푖푚 = 휟풕 풕−풕풐 훥푡→0 훥푡 풅풕 If an object changes its speed in a given time, there is acceleration. If the acceleration is constant, that is, if the object is increasing or decreasing its speed at a 'steady' rate, then we can use the formulas below. ex) The following snapshot is an object with constant acceleration. Notice that the object’s change in position increases at a steady rate t=0 t=1s t=2s t=3s ex) The following snapshot is an object when the acceleration is not constant. Notice that speed of the object speeds up and slows down erratically. t=0 t=1s t=2s t=3s t=4s t=5s t=6s The following formulas can only be used when the acceleration ‘a’ is constant !! 2 2 2 v = vo+ at x – xo = vot + (½)at v = vo +2a(x – xo) vo : initial velocity, v : final velocity, xo : initial position, x : final position ■ Warm-up Questions *~assume acceleration is constant for all questions 1. A car travels a distance of 400 meters in 20 seconds at constant speed. What is the speed of the car? (The car is moving at a straight line) (a) 6m/s (b) 12m/s (c) 16m/s (d) 20m/s 2. If you run at a constant speed of 5m/s for 5 seconds, how many meters did you cover? (Assume you are running in a straight line) (a) 25m (b) 16m (c) 12m (d) 3m 3. A runner travels 1.5 laps around circular track in a time of 50s. The diameter of the track is 40m and its circumference is 126m. i) Find the average speed of the runner (a) 2.55m/s (b) 3.78m/s (c) 5.25m/s (d) 7.43m/s ii) Find the runner’s average velocity (a) 3.52m/s (b) 1.25m/s (c) 0.80m/s (d) 0.21m/s 2 2 2 v = vo+ at x – xo = vot + (½)at v = vo +2a(x – xo) 4. A jogger starts from rest and speeds up with an acceleration of 3m/s2 in 2 seconds, what is the speed at 2 seconds? (a) 3m/s (b) 5m/s (c) 6m/s (d) 14m/s 5. Car-1 is moving at a constant speed of 30m/s. At the instant car-1 pass car-2, car-2 starts from rest and speeds up to 30m/s in 10 seconds. Which car traveled further in 10 seconds? 6. A body with initial velocity 8m/s moves along a straight line with constant acceleration and travels 640m in 40s. For the 40s interval, i) find the average velocity a) 16m/s b) 12m/s c) 6m/s d) 0.6m/s ii) find the final velocity, that is the velocity when the time is t=40s. (you need to find the acceleration first and then solve for final velocity) a) 29m/s b) 24m/s c) 19m/s d) 12m/s 7. A rocket-propelled car starts from rest and moves in a straight line with a=5m/s² for 8 seconds until the fuel is exhausted. It then continues with constant velocity. What distance does the rocket-car cover in 12 seconds? a) 160m b) 260m c) 320m d) 540m 2 2 2 v = vo+ at x – xo = vot + (½)at v = vo +2a(x – xo) 8. A box slides down an incline with uniform acceleration. It starts from rest and attains a speed of 2.7m/s in 3s. i) Find the acceleration a) 7.25m/s2 b) 4.51m/s2 c) 2.13m/s2 d) 0.90m/s2 ii) Find the distance moved in the first 6s a) 1.20m b) 5.9m c) 16.2m d) 21.8m Signs of the velocity and acceleration 'a' If a certain direction is chosen as positive, the sign of the velocity in that direction will be positive whether the object is speeding up or slowing down. If the object reverse direction, then the velocity will be negative whether the object is speeding up or slowing down. If ‘a’ is positive(a>0), then the acceleration is in the positive x-direction(right). If ‘a’ is negative(a<0), then the acceleration is in the negative x-direction(left). Determine the signs of ‘a’ for the following cases. We choose right as positive direction. (a) If an object is moving to the right and the speed is increasing (b) If an object is moving to the right and the speed is decreasing (c) If an object is moving to the left and the speed is increasing (d) If an object is moving to the left and the speed is decreasing 9. An object is moving with uniform(=constant) acceleration has a velocity of 12m/s in the positive x- direction when its x-coordinate is 3m. i) If its x-coordinate 2s later is –5m, what is its acceleration? a) 16m/s2 b) –16m/s2 c) 12m/s2 d) –12m/s2 ii) Find the velocity at 2s a) 14m/s b) –14m/s c) 20m/s d) –20m/s iii) Find the time when the object momentarily stops. a) 0.25s b) 0.50s c) 0.75s d) 1.00s .
Load more

Recommended publications
  • Rotational Motion (The Dynamics of a Rigid Body)
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body) Henry Semat City College of New York Robert Katz University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicskatz Part of the Physics Commons Semat, Henry and Katz, Robert, "Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body)" (1958). Robert Katz Publications. 141. https://digitalcommons.unl.edu/physicskatz/141 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert Katz Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 11 Rotational Motion (The Dynamics of a Rigid Body) 11-1 Motion about a Fixed Axis The motion of the flywheel of an engine and of a pulley on its axle are examples of an important type of motion of a rigid body, that of the motion of rotation about a fixed axis. Consider the motion of a uniform disk rotat­ ing about a fixed axis passing through its center of gravity C perpendicular to the face of the disk, as shown in Figure 11-1. The motion of this disk may be de­ scribed in terms of the motions of each of its individual particles, but a better way to describe the motion is in terms of the angle through which the disk rotates.
    [Show full text]
  • Kinematics Study of Motion
    Kinematics Study of motion Kinematics is the branch of physics that describes the motion of objects, but it is not interested in its causes. Itziar Izurieta (2018 october) Index: 1. What is motion? ............................................................................................ 1 1.1. Relativity of motion ................................................................................................................................ 1 1.2.Frame of reference: Cartesian coordinate system ....................................................................................................................................................................... 1 1.3. Position and trajectory .......................................................................................................................... 2 1.4.Travelled distance and displacement ....................................................................................................................................................................... 3 2. Quantities of motion: Speed and velocity .............................................. 4 2.1. Average and instantaneous speed ............................................................ 4 2.2. Average and instantaneous velocity ........................................................ 7 3. Uniform linear motion ................................................................................. 9 3.1. Distance-time graph .................................................................................. 10 3.2. Velocity-time
    [Show full text]
  • Introduction to Robotics Lecture Note 5: Velocity of a Rigid Body
    ECE5463: Introduction to Robotics Lecture Note 5: Velocity of a Rigid Body Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 5 (ECE5463 Sp18) Wei Zhang(OSU) 1 / 24 Outline Introduction • Rotational Velocity • Change of Reference Frame for Twist (Adjoint Map) • Rigid Body Velocity • Outline Lecture 5 (ECE5463 Sp18) Wei Zhang(OSU) 2 / 24 Introduction For a moving particle with coordinate p(t) 3 at time t, its (linear) velocity • R is simply p_(t) 2 A moving rigid body consists of infinitely many particles, all of which may • have different velocities. What is the velocity of the rigid body? Let T (t) represent the configuration of a moving rigid body at time t.A • point p on the rigid body with (homogeneous) coordinate p~b(t) and p~s(t) in body and space frames: p~ (t) p~ ; p~ (t) = T (t)~p b ≡ b s b Introduction Lecture 5 (ECE5463 Sp18) Wei Zhang(OSU) 3 / 24 Introduction Velocity of p is d p~ (t) = T_ (t)p • dt s b T_ (t) is not a good representation of the velocity of rigid body • - There can be 12 nonzero entries for T_ . - May change over time even when the body is under a constant velocity motion (constant rotation + constant linear motion) Our goal is to find effective ways to represent the rigid body velocity. • Introduction Lecture 5 (ECE5463 Sp18) Wei Zhang(OSU) 4 / 24 Outline Introduction • Rotational Velocity • Change of Reference Frame for Twist (Adjoint Map) • Rigid Body Velocity • Rotational Velocity Lecture 5 (ECE5463 Sp18) Wei Zhang(OSU) 5 / 24 Illustrating
    [Show full text]
  • Unit 1: Motion
    Macomb Intermediate School District High School Science Power Standards Document Physics The Michigan High School Science Content Expectations establish what every student is expected to know and be able to do by the end of high school. They also outline the parameters for receiving high school credit as dictated by state law. To aid teachers and administrators in meeting these expectations the Macomb ISD has undertaken the task of identifying those content expectations which can be considered power standards. The critical characteristics1 for selecting a power standard are: • Endurance – knowledge and skills of value beyond a single test date. • Leverage - knowledge and skills that will be of value in multiple disciplines. • Readiness - knowledge and skills necessary for the next level of learning. The selection of power standards is not intended to relieve teachers of the responsibility for teaching all content expectations. Rather, it gives the school district a common focus and acts as a safety net of standards that all students must learn prior to leaving their current level. The following document utilizes the unit design including the big ideas and real world contexts, as developed in the science companion documents for the Michigan High School Science Content Expectations. 1 Dr. Douglas Reeves, Center for Performance Assessment Unit 1: Motion Big Ideas The motion of an object may be described using a) motion diagrams, b) data, c) graphs, and d) mathematical functions. Conceptual Understandings A comparison can be made of the motion of a person attempting to walk at a constant velocity down a sidewalk to the motion of a person attempting to walk in a straight line with a constant acceleration.
    [Show full text]
  • Newton Euler Equations of Motion Examples
    Newton Euler Equations Of Motion Examples Alto and onymous Antonino often interloping some obligations excursively or outstrikes sunward. Pasteboard and Sarmatia Kincaid never flits his redwood! Potatory and larboard Leighton never roller-skating otherwhile when Trip notarizes his counterproofs. Velocity thus resulting in the tumbling motion of rigid bodies. Equations of motion Euler-Lagrange Newton-Euler Equations of motion. Motion of examples of experiments that a random walker uses cookies. Forces by each other two examples of example are second kind, we will refer to specify any parameter in. 213 Translational and Rotational Equations of Motion. Robotics Lecture Dynamics. Independence from a thorough description and angular velocity as expected or tofollowa userdefined behaviour does it only be loaded geometry in an appropriate cuts in. An interface to derive a particular instance: divide and author provides a positive moment is to express to output side can be run at all previous step. The analysis of rotational motions which make necessary to decide whether rotations are. For xddot and whatnot in which a very much easier in which together or arena where to use them in two backwards operation complies with respect to rotations. Which influence of examples are true, is due to independent coordinates. On sameor adjacent joints at each moment equation is also be more specific white ellipses represent rotations are unconditionally stable, for motion break down direction. Unit quaternions or Euler parameters are known to be well suited for the. The angular momentum and time and runnable python code. The example will be run physics examples are models can be symbolic generator runs faster rotation kinetic energy.
    [Show full text]
  • Rotation: Moment of Inertia and Torque
    Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn that where the force is applied and how the force is applied is just as important as how much force is applied when we want to make something rotate. This tutorial discusses the dynamics of an object rotating about a fixed axis and introduces the concepts of torque and moment of inertia. These concepts allows us to get a better understanding of why pushing a door towards its hinges is not very a very effective way to make it open, why using a longer wrench makes it easier to loosen a tight bolt, etc. This module begins by looking at the kinetic energy of rotation and by defining a quantity known as the moment of inertia which is the rotational analog of mass. Then it proceeds to discuss the quantity called torque which is the rotational analog of force and is the physical quantity that is required to changed an object's state of rotational motion. Moment of Inertia Kinetic Energy of Rotation Consider a rigid object rotating about a fixed axis at a certain angular velocity. Since every particle in the object is moving, every particle has kinetic energy. To find the total kinetic energy related to the rotation of the body, the sum of the kinetic energy of every particle due to the rotational motion is taken. The total kinetic energy can be expressed as ..
    [Show full text]
  • Lecture 24 Angular Momentum
    LECTURE 24 ANGULAR MOMENTUM Instructor: Kazumi Tolich Lecture 24 2 ¨ Reading chapter 11-6 ¤ Angular momentum n Angular momentum about an axis n Newton’s 2nd law for rotational motion Angular momentum of an rotating object 3 ¨ An object with a moment of inertia of � about an axis rotates with an angular speed of � about the same axis has an angular momentum, �, given by � = �� ¤ This is analogous to linear momentum: � = �� Angular momentum in general 4 ¨ Angular momentum of a point particle about an axis is defined by � � = �� sin � = ��� sin � = �-� = ��. � �- ¤ �⃗: position vector for the particle from the axis. ¤ �: linear momentum of the particle: � = �� �⃗ ¤ � is moment arm, or sometimes called “perpendicular . Axis distance.” �. Quiz: 1 5 ¨ A particle is traveling in straight line path as shown in Case A and Case B. In which case(s) does the blue particle have non-zero angular momentum about the axis indicated by the red cross? A. Only Case A Case A B. Only Case B C. Neither D. Both Case B Quiz: 24-1 answer 6 ¨ Only Case A ¨ For a particle to have angular momentum about an axis, it does not have to be Case A moving in a circle. ¨ The particle can be moving in a straight path. Case B ¨ For it to have a non-zero angular momentum, its line of path is displaced from the axis about which the angular momentum is calculated. ¨ An object moving in a straight line that does not go through the axis of rotation has an angular position that changes with time. So, this object has an angular momentum.
    [Show full text]
  • Rotational Motion and Angular Momentum 317
    CHAPTER 10 | ROTATIONAL MOTION AND ANGULAR MOMENTUM 317 10 ROTATIONAL MOTION AND ANGULAR MOMENTUM Figure 10.1 The mention of a tornado conjures up images of raw destructive power. Tornadoes blow houses away as if they were made of paper and have been known to pierce tree trunks with pieces of straw. They descend from clouds in funnel-like shapes that spin violently, particularly at the bottom where they are most narrow, producing winds as high as 500 km/h. (credit: Daphne Zaras, U.S. National Oceanic and Atmospheric Administration) Learning Objectives 10.1. Angular Acceleration • Describe uniform circular motion. • Explain non-uniform circular motion. • Calculate angular acceleration of an object. • Observe the link between linear and angular acceleration. 10.2. Kinematics of Rotational Motion • Observe the kinematics of rotational motion. • Derive rotational kinematic equations. • Evaluate problem solving strategies for rotational kinematics. 10.3. Dynamics of Rotational Motion: Rotational Inertia • Understand the relationship between force, mass and acceleration. • Study the turning effect of force. • Study the analogy between force and torque, mass and moment of inertia, and linear acceleration and angular acceleration. 10.4. Rotational Kinetic Energy: Work and Energy Revisited • Derive the equation for rotational work. • Calculate rotational kinetic energy. • Demonstrate the Law of Conservation of Energy. 10.5. Angular Momentum and Its Conservation • Understand the analogy between angular momentum and linear momentum. • Observe the relationship between torque and angular momentum. • Apply the law of conservation of angular momentum. 10.6. Collisions of Extended Bodies in Two Dimensions • Observe collisions of extended bodies in two dimensions. • Examine collision at the point of percussion.
    [Show full text]
  • Shape Rotational Inertia, I I = M1r + M2r + ... (Point Masses) I = ∫ R Dm
    Chs. 10 - Rotation & Angular Motion Dan Finkenstadt, fi[email protected] October 26, 2015 Shape Rotational Inertia, I I one aspect of shape is the center of mass I How difficult is it to spin an object about (c.o.m.) any axis? I another is how mass is distributed about the I Formulas: (distance r from axis) c.o.m. 2 2 I = m1r1 + m2r2 + ::: I an important parameter describing shape is called, (point masses) 1. Rotational Inertia (in our book) Z 2. Moment of Inertia (in most other books) I = r2 dm (solid object) 3.2 nd Moment of Mass (in engineering, math books) Solution: 8 kg(2 m)2 + 5 kg(8 m2) + 6 kg(2 m)2 = 96 kg · m2 P 2 Rotational Inertia: I = miri Two you should memorize: Let's calculate one of these: I mass M concentrated at radius R I = MR2 I Disk of mass M, radius R, about center 1 I = MR2 solid disk 2 Parallel Axis Theorem Active Learning Exercise Problem: Rotational Inertia of Point We can calculate the rotational inertia about an Masses axis parallel to the c.o.m. axis Four point masses are located at the corners of a square with sides 2.0 m, 2 I = Icom + m × shift as shown. The 1.0 kg mass is at the origin. What is the rotational inertia of the four-mass system about an I Where M is the total mass of the body axis passing through the mass at the origin and pointing out of the screen I And shift is the distance between the axes (page)? I REMEMBER: The axes must be parallel! What is so useful about I? Rotation I radians! It connects up linear quantities to angular I right-hand rule quantities, e.g., kinetic energy angular speed v ! = rad r s I v = r! 2 2 I ac = r! 1 2 1 2v 1 2 mv ! mr ! I! ang.
    [Show full text]
  • 8.01 Classical Mechanics Chapter 21
    Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis 21.1 Introduction ........................................................................................................... 1 21.2 Translational Equation of Motion ....................................................................... 1 21.3 Translational and Rotational Equations of Motion ........................................... 1 21.3.1 Summary ......................................................................................................... 6 21.4 Translation and Rotation of a Rigid Body Undergoing Fixed Axis Rotation . 6 21.5 Work-Energy Theorem ........................................................................................ 7 21.6 Worked Examples ................................................................................................. 8 Example 21.1 Angular Impulse ............................................................................... 8 Example 21.2 Person on a railroad car moving in a circle .................................... 9 Example 21.3 Torque, Rotation and Translation: Yo-Yo ................................... 13 Example 21.4 Cylinder Rolling Down Inclined Plane ......................................... 16 Example 21.5 Bowling Ball ..................................................................................... 21 Example 21.6 Rotation and Translation Object and Stick Collision ................. 25 Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis Accordingly, we find Euler and D'Alembert
    [Show full text]
  • Circular Motion
    Circular motion Aug. 22, 2017 Until now, we have been observers to Newtonian physics through inertial reference frames. From our discussion of Newton's laws, these are frames which obey Newton's first law{they don't accelerate and therefore move at constant velocity. Here the rules of vector analysis apply and we can change reference frames so that the frame is not moving. In the context of momentum conservation, we saw that particularly nice inertial frames are those in which one object isn't moving and the frame in which the center of mass is not moving. However, we regularly encounter situations where inertial frames don't apply{for instance when we accelerate. Recall that the acceleration is the change in velocity with respect to time, d~v ~a = : (1) dt The change in velocity can be a change in magnitude, as we saw in one dimensional motion, but it also may be a change in direction with no change in magnitude. This occurs in circular motion. Let's see how this works! A circular worldline for a particle can be written ~x(t) = R(cos(2πt=T ); sin(2πt=T ); 0): (2) Every ∆t = T , the particle returns to its starting point. The velocity of this worldline is 2πR ~v(t) = (− sin(2πt=T ); cos(2πt=T ); 0); (3) T and its acceleration is 4π2R 4π2 ~a(t) = (− cos(2πt=T ); − sin(2πt=T ); 0) = − ~x(t): (4) T 2 T 2 1 Notice that the last equation is the same differential equation as we saw in the static friction case and the spring{hmm, I wonder if there is a differential equation that describes periodic motion.
    [Show full text]
  • Linear Motion
    Linear Motion OBJECTIVES • to compare the relationship between position vs time and velocity vs time for different types of linear motion • to learn to find the characteristics of linear motion from a position vs time graph and a velocity vs time graph • to measure the value of gravitational acceleration g on Earth by the free fall method and compare it to the accepted value EQUIPMENT battery powered constant velocity buggy masking tape cardboard PASCO motion sensor sail cart track small rubber object and whiffle ball stopwatch 30-m measuring tape picket fence with 5-cm band spacing wide photogate on stand floor mat c 2012 Advanced Instructional Systems Inc. and Arizona State University Department of Physics 1 Figure 1 INTRODUCTION AND THEORY Kinematics is the area of physics that studies types of motion without specifying what caused it. The three major types of simple linear motion are constant velocity motion, uniformly accelerated linear motion, and free fall. The basic physics quantities used to describe the motion of an object are: position, distance, displacement, speed, velocity, and acceleration. The description is sufficient if one can specify the location of the object at all times. Position is simply the distance from an arbitrary chosen point called the point of reference. The point of reference is usually aligned with the origin on the coordinate axis. For one dimensional motion (or motion in a straight line), the position at any time can be specified by a single value, x or y, with units of distance. Positions to the right of the origin will be given a positive sign; positions to the left will be negative.
    [Show full text]