Analysis of Some Dynamic Tests II
|
|
- Cori Lynch
- 5 years ago
- Views:
Transcription
1 Analysis of Some Dynamic Tests II Introduction This study is based on test data provided by KuLTu. The data were obtained from dynamic and isometric tests with HUR Leg Extension/Curl Research Line machine and software. The test group Kalpa encompassed 27 male subjects. Of the dynamic (extension) tests we had to drop one test for left leg and two tests for right leg because of spurious data. The dynamic tests consisted of 13 kicks at increasing load from 2 to 8 bar. The Force-Velocity relationship is studied with reference to the Hill modell. Analysis using A- and B-parameters For dynamic tests the torque and angular data are collected from which the software calculates maximum torque (MTQ), maximun angular velocity (VEL), and power at maximum torque for each kick (POW). Naturally we expect the velocity to decrease with increasing load. In the following graph we have plotted the MTQ-VEL data for 25 tests (right leg) MTq i AVel i Fig. 1. Maximum torque (Nm) vs maximum angular velocity (deg/s) for 25 tests (right leg, extension). The graph suggests we may try to fit the individual MTQ-VEL data to linear relationship 1
2 (1) MTQ = A $ VEL + B The A-B parameters can be calculated for each test and then plotted in an A-B graph: LA j rla j LB j 39.2 Fig. 2. The A-B parameters for the 25 tests of fig. 1 are plotted in an A-B diagram. Also the best linear fit curve is indicated. As can been seen the A-B parameters correlate fairly well with each other. In fact, their Pearson correlation coefficient is -.8. The linear best fit in fig. 2 is given by (2) A =.238 $ B bar data The relation (1) further suggests that the ratio - B/A could be related to maximum velocity at small load. Indeed, if take the maximum velocity for each test and plot it with - B/A on the same graph we obtain the following picture. 2
3 maxv j 7 LB j LA j 6 5 mv j 24 Fig. 3. Maximum velocity for each test (maxv) plotted with the quotients - B/A for the tests. From the figure one can see that there is an obvious covariance (.87). The best linear fit predicts the maximum velocity from - B/A to within 1-15 % maxv j rvel j rvel j Fig. 4. Maximum velocity of each test (point) and the prediction based on - B/A (line). The linear prediction (regression) is given in this case by (3) MAXVEL =.554 $ B A (velocity in deg/s). Similarily we may expect the maximum power (MPOW) of each test to be correlated with 3
4 (4) P = B2 4 $ A $ 18 (in Watt). This is demonstrated by the following picture (Pearson s correlation =.932) P j 1 mpw j j 24 Fig. 5. Plotting of maximum power (mpw) and P calculated from equ (4). Both from experimental and theoretical points of view there are no reasons to expect a linear Force-Velocity curve for the whole range of loads. The situation is schematically represented in fig. 6. Nm M B1 B2 deg/s Fig. 6. If the schematic Force-Velocity curve is like this (concave), then the B-parameter should always be less then the isometric maximum (M ). 4
5 We can use our experimental data to investigate whether we have a situation like the one depicted in fig. 6. For this we make two selections of the data: (1) kicks for loads 5-8 bar and (2) kicks for loads 2-5 bar, and then calculate the corresponding A - B parameters MR j B j j 24 Fig. 7. Isometric maximum/right (MR) and the B-parameter for the test selection 1 (5-8 bar). As we can see from fig. 7 the B-parameter tends to be lower than the isometric maximum (extension) for the 25 subjects with a few exceptions. For the averages we have: mean(b) = 33.9 Nm, and mean(mr) = Nm. Also to be noted is that the power estimates calculated according to equ (4) are bad when based on the 5-8 bar data (corr(p, mpw) =.65) A j ra j B j Fig. 8. The A - B parameters for selection 1 (5-8 bar). For the selection 1 (5-8 bar) the linear A - B regression curve is given by 5
6 (5) A =.324 $ B +.54 corr(a,b) =.881 selection 1 (5-8 bar) The result for the selection 2 (2-5 bar) is: (6) A =.216 $ B corr(a,b) =.833 selection 2 (2-5 bar) For this selection we have B2 = mean(b) = 28.1 Nm. In neither case was there any significant correlation between the B-parameter and the mass of the subjects (corr(b1, M) =.4; corr(b2, M) = -.4), or between isometric maximum torque (extension) and the mass of the subjects (corr. =.23). The later may seem paradoxical, but is likely due to the small variation of mass and length in the group: n = 25 Max Min Mean StDeviation Length (cm) Mass (kg) Age (year) Above we have presented data for the right leg. For the left leg the data seem to be more spread (lower values for the the pertinent correlation coefficients), which may e.g. be due to fatigue (the right leg was tested first). This could be checked by testing with the left leg first. Anyway, comparing A- and B-parameters from different test is only meaningful if based on data from exercises in the same load range. Hill modell For the simple Hill modell [1] one has a relation of the form (7) F = F b av b + v between the maximal force F of a muscle and its velocity of contraction, v. Here a and b are parameters for the muscle, and F is maximal force at v = (isometric). 6
7 Hill modell Force (N) F( v) v 1 Velocity (cm/s) Fig. 9. Typical Force-Velocity curve for the Hill modell. The maximum contraction velocity (v ) for F = is according to (7) given by (8) v = b a F Experiments with skeletal muscles show that (9) a F = b v l.25 With this assumption one can calculate the maximum power P, using P = Fv and (9), to be at (1) P =.95 $ F $ v v max P =.31 $ v F max P =.31 $ F where and v maxp and F maxp indicate the velocity and force at maximum power. For our test data we can compare maximum power for each subject with the product of maximum torque (isometric test) and maximum veclocity in the test (which here really is not for zero load but for 2 bar load plus the weight of the leg). 7
8 mpw j 9 rmp j mp j Fig. 1. Measured maximum power (mpw) and the best linear fit (rmp) based on the product (mp) of maximum torque (isometric) and maximum velocity (2 bar load). In fig. 1 the product mp was calculated as (11) mp = MTQ $ VEL $ 18 (MTQ, maximum isometric torque; VEL, maximum velocity (deg/s) for dynamic extension). The correlation coefficient with actual measured maximal power mpw was.91, the linear best fit given by (12) mpw =.385 $ mp (Watt) This result is not directly comparable with the Hill modell because we could not use the maximum velocity for zero load (however the slope-coefficient in (12) is consistent with the prediction (1) that maximum power is attained at around one third of maximum isometric torque). This could e.g. be indicated by the nonzero intercept in (12). The Hill equation can also be written on the forms (13) 1 F F v = v 1 + c $ F F 1 v v F = F 1 + c $ v v where c is the shape parameter ( related to a by c = F /a ). In terms of the dimensionless variables the Hill curve is thus determined by a single parameter c. If we assume (13) to be valid then the shape parameter c is related to the force F pmax at maximum power through 8
9 (14) c = 1 2 $ F pmax F F pmax F 2 F p max F = 1 c 1 + c 1 which can be calculated from the measurement data. Test nr Mean Max Min StDeviation c - parameter correlation slope intercept The above table summarizes the results (right leg). The first column gives the shape parameter calculated for each test. The next column gives the correlation between actual measured velocities (VEL) and the ones computed from Hill s equation using the calculated shape parameter (VELh). The slope (α) and intercept (β) describe the best linear fit curve 9
10 (15) VEL = $ VELh + The high correlations above are not spectacular; we get the same kind of figures even if we compute correlations between data from different tests (different persons). This is not surprising if the the data complies with the Hill modell, because then the data should follow the same simple pattern. Indeed, if we map all the data as in fig. 1 but use dimensionless variables (16) f = TQ TQ isomax V = VEL VEL (here TQ isomax is the isometric maximum, and VEL the maximum velocity in the test with least load; i.e. 2 bar) the we get the figure: MTq i.6 MR f( i) AVel i VEL < f( i) > Fig. 11. The same plot as in fig. 1 but with dimensionless variables (16) instead ( f along the vertical, V along the horizontal axis). Fig. 11 emphasizes the common systematic trend in the data which also explains the high intercorrelation between different tests. One interesting feature of the Hill modell is the symmetrical way it treats force and velocity as can be seen from equ (13) and from its geometrical shape if plotted using F/F and v/v. 1
11 F/F Max power fpmax vpmax V/V Fig. 12. The symmetrical Hill curve. The parameter c describes the deviation from a straight line. The maximum power ( P = F V ) is attained at the midpoint of the curve. The corresponding fractional velocities and forces will according to the Hill curve be equal at this point. According to the Hill equation the maximum power will be attained at the midpoint of the curve where F pmax /F = v pmax /v. Now we can compute F pmax /F for each test from the data. n = 25 Mean Max Min StDeviation Fpmax/F Interestingly, the mean value.35 is quite close to the value.31 predicted by (1) based on a shape parameter equal to 4. Thus, the maximum power is attained at about one third of the maximum force and the maximum velocity. From this one could estimate the maximum velocity v which may be difficult to measure directly. Also, it can be determined from the differential quotient of velocity and force at the point of maximum power according to (17) v = F dv df pmax From the data we compute the average for v pmax to be 41 grad/s which would correpsond to an average value 41/.35 = 117 grad/s (2.4 rad/s) for v. (If we take the moment radius of the quadriceps muscle to be 3.3 cm [1] then 2 rad/s corresponds to a muscle contraction velocity of cm/s =.66 m/s.) 11
12 An inspection of the data also reveals that the tests are concentrated at the left side of the Hill curve; i.e. the low velocity/heavy load end. F/F Test data concentrated here fpmax vpmax V/V Fig. 13. Test are concentrated at the low velocity/high load end. We will also study the connection between the Hill modell and the ( A, B ) - parameters discussed in the first part of this paper. If the Hill modell describes the data then the ( A, B ) - parameters should be approximated by the parameters decribing the tangent line to the Hill curve touching it at a point in the middle of the measurement range. This leads to the following expression for the parameters (18) A = F v B = F + F v 1 + c 1+ c v v c 1+ c v v v 2 where F and v are in the middle of the measurement range. Equ (18) predicts a linear relationship between A and B of the sort (19) A = 1 v $ B + F v where the slope and the intercept are thus given by 12
13 (2) = 1 v = F v If we compute the average of 1/VEL and TQ/VEL for the 6th kick (of 13 in all) of each test for the whole data we get mean(1/vel) =.298, mean(tq/vel) =.425, to be compared with values.238 and.258 given by equ (2). (Interestingly, if we compute the same averages for the 3d kicks instead we get the values.238 resp..255 which match the regression coefficients (2) very well!) This cursory comparision seems to show that the Hill modell can explain a number of important features of the data. Discussion It seems that the Hill modell could be of help interpreting the data even if the data is not optimal for a comparision in this case. Now we used only the maximum velocity for each kick, but calculating the velocity at maximum power would be more useful (in this case these are not too far off from each other, though). This calculation could be added to the HUR software. In the Hill modell one can also incorporate the muscle length factor which means that the constant maximum isometric force F in (13) is to be replaced by one that varies with the length of the muscle (or the joint angle). The length factor can be studied experimentally by measuring the maximum isometric force for a range of joint angles between 9 and 18 degree. This factor must furthermore include the geometry of the muscle and the leg which relate the net muscular force with net muscular torque. These things, and the effect of inertia, will be also studied in a simulation in order to assess their importance for analyzing the data. In the theory the shape parameter is a measure how fast the muscle is. The larger the shape parameter the more fast fibers in the muscle. This could perhaps be tested by also taking the time for a 6 meter sprinting of the test subjects (in case of fairly well trained subjects). References: 1. Nigg, B M, Herzog W: Biomechanics of the Musculo-skeletal System, Wiley Keener J, Sneyd J: Mathematical Physiology, Springer Frank Borg (borgbros@netti.fi) 13
5. Find the slope intercept equation of the line parallel to y = 3x + 1 through the point (4, 5).
Rewrite using rational eponents. 2 1. 2. 5 5. 8 4 4. 4 5. Find the slope intercept equation of the line parallel to y = + 1 through the point (4, 5). 6. Use the limit definition to find the derivative
More informationPhysics 6A Lab Experiment 6
Biceps Muscle Model Physics 6A Lab Experiment 6 Introduction This lab will begin with some warm-up exercises to familiarize yourself with the theory, as well as the experimental setup. Then you ll move
More informationPhysics 6A Lab Experiment 6
Rewritten Biceps Lab Introduction This lab will be different from the others you ve done so far. First, we ll have some warmup exercises to familiarize yourself with some of the theory, as well as the
More informationAngular Motion Maximum Hand, Foot, or Equipment Linear Speed
Motion Maximum Hand, Foot, or Equipment Linear Speed Biomechanical Model: Mo3on Maximum Hand, Foot, or Equipment Linear Speed Hand, Foot, or Equipment Linear Speed Sum of Joint Linear Speeds Principle
More informationChapter 8 continued. Rotational Dynamics
Chapter 8 continued Rotational Dynamics 8.4 Rotational Work and Energy Work to accelerate a mass rotating it by angle φ F W = F(cosθ)x x = s = rφ = Frφ Fr = τ (torque) = τφ r φ s F to s θ = 0 DEFINITION
More informationEquilibrium. For an object to remain in equilibrium, two conditions must be met. The object must have no net force: and no net torque:
Equilibrium For an object to remain in equilibrium, two conditions must be met. The object must have no net force: F v = 0 and no net torque: v τ = 0 Worksheet A uniform rod with a length L and a mass
More informationChapter 9. Rotational Dynamics
Chapter 9 Rotational Dynamics 9.1 The Action of Forces and Torques on Rigid Objects In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination
More informationWeek 8: Correlation and Regression
Health Sciences M.Sc. Programme Applied Biostatistics Week 8: Correlation and Regression The correlation coefficient Correlation coefficients are used to measure the strength of the relationship or association
More informationChapter 9. Rotational Dynamics
Chapter 9 Rotational Dynamics 9.1 The Action of Forces and Torques on Rigid Objects In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination
More informationLab 11: Rotational Dynamics
Lab 11: Rotational Dynamics Objectives: To understand the relationship between net torque and angular acceleration. To understand the concept of the moment of inertia. To understand the concept of angular
More informationBasic Biomechanics II DEA 325/651 Professor Alan Hedge
Basic Biomechanics II DEA 325/651 Professor Alan Hedge Definitions! Scalar quantity quantity with magnitude only (e.g. length, weight)! Vector quantity quantity with magnitude + direction (e.g. lifting
More informationChapter 8 continued. Rotational Dynamics
Chapter 8 continued Rotational Dynamics 8.4 Rotational Work and Energy Work to accelerate a mass rotating it by angle φ F W = F(cosθ)x x = rφ = Frφ Fr = τ (torque) = τφ r φ s F to x θ = 0 DEFINITION OF
More informationRotation of Rigid Objects
Notes 12 Rotation and Extended Objects Page 1 Rotation of Rigid Objects Real objects have "extent". The mass is spread out over discrete or continuous positions. THERE IS A DISTRIBUTION OF MASS TO "AN
More informationPhysics 2A Chapter 10 - Rotational Motion Fall 2018
Physics A Chapter 10 - Rotational Motion Fall 018 These notes are five pages. A quick summary: The concepts of rotational motion are a direct mirror image of the same concepts in linear motion. Follow
More informationRotational Dynamics continued
Chapter 9 Rotational Dynamics continued 9.1 The Action of Forces and Torques on Rigid Objects Chapter 8 developed the concepts of angular motion. θ : angles and radian measure for angular variables ω :
More informationMechanics II. Which of the following relations among the forces W, k, N, and F must be true?
Mechanics II 1. By applying a force F on a block, a person pulls a block along a rough surface at constant velocity v (see Figure below; directions, but not necessarily magnitudes, are indicated). Which
More informationProblem Set 1: Solutions 2
UNIVERSITY OF ALABAMA Department of Physics and Astronomy PH 125 / LeClair Spring 2009 Problems due 15 January 2009. Problem Set 1: Solutions 2 1. A person walks in the following pattern: 3.1 km north,
More informationExperiment #9 Comments, Thoughts and Suggestions
Experiment #9 Comments, Thoughts and Suggestions The purpose of this paper is to provide you with some information which may be useful for solving the pre-lab questions and performing the lab. I will attempt
More informationFundamentals Physics. Chapter 10 Rotation
Fundamentals Physics Tenth Edition Halliday Chapter 10 Rotation 10-1 Rotational Variables (1 of 15) Learning Objectives 10.01 Identify that if all parts of a body rotate around a fixed axis locked together,
More informationANGULAR KINETICS (Part 1 Statics) Readings: McGinnis (2005), Chapter 5.
NGUL KINTICS (Part 1 Statics) eadings: McGinnis (2005), Chapter 5. 1 Moment of Force or Torque: What causes a change in the state of linear motion of an object? Net force ( F = ma) What causes a change
More informationChapter 4. Oscillatory Motion. 4.1 The Important Stuff Simple Harmonic Motion
Chapter 4 Oscillatory Motion 4.1 The Important Stuff 4.1.1 Simple Harmonic Motion In this chapter we consider systems which have a motion which repeats itself in time, that is, it is periodic. In particular
More informationEQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS (Section 17.4) Today s Objectives: Students will be able to analyze the planar kinetics of a rigid
EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS (Section 17.4) Today s Objectives: Students will be able to analyze the planar kinetics of a rigid body undergoing rotational motion. APPLICATIONS The crank
More informationWorksheet for Exploration 10.1: Constant Angular Velocity Equation
Worksheet for Exploration 10.1: Constant Angular Velocity Equation By now you have seen the equation: θ = θ 0 + ω 0 *t. Perhaps you have even derived it for yourself. But what does it really mean for the
More informationRotational Kinematics and Dynamics. UCVTS AIT Physics
Rotational Kinematics and Dynamics UCVTS AIT Physics Angular Position Axis of rotation is the center of the disc Choose a fixed reference line Point P is at a fixed distance r from the origin Angular Position,
More informationSample Questions to the Final Exam in Math 1111 Chapter 2 Section 2.1: Basics of Functions and Their Graphs
Sample Questions to the Final Eam in Math 1111 Chapter Section.1: Basics of Functions and Their Graphs 1. Find the range of the function: y 16. a.[-4,4] b.(, 4],[4, ) c.[0, ) d.(, ) e.. Find the domain
More informationRotational Dynamics Smart Pulley
Rotational Dynamics Smart Pulley The motion of the flywheel of a steam engine, an airplane propeller, and any rotating wheel are examples of a very important type of motion called rotational motion. If
More informationRotational Motion. Figure 1: Torsional harmonic oscillator. The locations of the rotor and fiber are indicated.
Rotational Motion 1 Purpose The main purpose of this laboratory is to familiarize you with the use of the Torsional Harmonic Oscillator (THO) that will be the subject of the final lab of the course on
More information2010 F=ma Solutions. that is
2010 F=ma Solutions 1. The slope of a position vs time graph gives the velocity of the object So you can see that the position from B to D gives the steepest slope, so the speed is the greatest in that
More informationAngular Motion. Experiment 4
Experiment 4 Angular Motion Before starting the experiment, you need to be familiar with the concept of angular position θ, angular velocity ω, angular acceleration α, torque τ, moment of inertia I. See
More informationChapter 8. Rotational Equilibrium and Rotational Dynamics. 1. Torque. 2. Torque and Equilibrium. 3. Center of Mass and Center of Gravity
Chapter 8 Rotational Equilibrium and Rotational Dynamics 1. Torque 2. Torque and Equilibrium 3. Center of Mass and Center of Gravity 4. Torque and angular acceleration 5. Rotational Kinetic energy 6. Angular
More informationThis is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail.
This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. uthor(s): Rahikainen, hti; Virmavirta, Mikko Title: Constant Power
More information( )( ) ( )( ) Fall 2017 PHYS 131 Week 9 Recitation: Chapter 9: 5, 10, 12, 13, 31, 34
Fall 07 PHYS 3 Chapter 9: 5, 0,, 3, 3, 34 5. ssm The drawing shows a jet engine suspended beneath the wing of an airplane. The weight W of the engine is 0 00 N and acts as shown in the drawing. In flight
More informationJumping on a scale. Data acquisition (TI 83/TI84)
Jumping on a scale Data acquisition (TI 83/TI84) Objective: In this experiment our objective is to study the forces acting on a scale when a person jumps on it. The scale is a force probe connected to
More informationRotation of Rigid Objects
Notes 12 Rotation and Extended Objects Page 1 Rotation of Rigid Objects Real objects have "extent". The mass is spread out over discrete or continuous positions. THERE IS A DISTRIBUTION OF MASS TO "AN
More informationt = g = 10 m/s 2 = 2 s T = 2π g
Annotated Answers to the 1984 AP Physics C Mechanics Multiple Choice 1. D. Torque is the rotational analogue of force; F net = ma corresponds to τ net = Iα. 2. C. The horizontal speed does not affect the
More informationChapter 8 Rotational Motion
Chapter 8 Rotational Motion Chapter 8 Rotational Motion In this chapter you will: Learn how to describe and measure rotational motion. Learn how torque changes rotational velocity. Explore factors that
More informationBiomechanical Modelling of Musculoskeletal Systems
Biomechanical Modelling of Musculoskeletal Systems Lecture 6 Presented by Phillip Tran AMME4981/9981 Semester 1, 2016 The University of Sydney Slide 1 The Musculoskeletal System The University of Sydney
More informationDaily WeBWorK. 1. Below is the graph of the derivative f (x) of a function defined on the interval (0, 8).
Daily WeBWorK 1. Below is the graph of the derivative f (x) of a function defined on the interval (0, 8). (a) On what intervals is f (x) concave down? f (x) is concave down where f (x) is decreasing, so
More informationRotation. Kinematics Rigid Bodies Kinetic Energy. Torque Rolling. featuring moments of Inertia
Rotation Kinematics Rigid Bodies Kinetic Energy featuring moments of Inertia Torque Rolling Angular Motion We think about rotation in the same basic way we do about linear motion How far does it go? How
More informationHomework 4 Solutions, 2/2/7
Homework 4 Solutions, 2/2/7 Question Given that the number a is such that the following limit L exists, determine a and L: x 3 a L x 3 x 2 7x + 2. We notice that the denominator x 2 7x + 2 factorizes as
More informationAP Physics 1- Torque, Rotational Inertia, and Angular Momentum Practice Problems FACT: The center of mass of a system of objects obeys Newton s second law- F = Ma cm. Usually the location of the center
More informationChapter 9: Rotational Dynamics Tuesday, September 17, 2013
Chapter 9: Rotational Dynamics Tuesday, September 17, 2013 10:00 PM The fundamental idea of Newtonian dynamics is that "things happen for a reason;" to be more specific, there is no need to explain rest
More informationChapter 8 continued. Rotational Dynamics
Chapter 8 continued Rotational Dynamics 8.6 The Action of Forces and Torques on Rigid Objects Chapter 8 developed the concepts of angular motion. θ : angles and radian measure for angular variables ω :
More informationPhysics. Chapter 8 Rotational Motion
Physics Chapter 8 Rotational Motion Circular Motion Tangential Speed The linear speed of something moving along a circular path. Symbol is the usual v and units are m/s Rotational Speed Number of revolutions
More informationDefinition. is a measure of how much a force acting on an object causes that object to rotate, symbol is, (Greek letter tau)
Torque Definition is a measure of how much a force acting on an object causes that object to rotate, symbol is, (Greek letter tau) = r F = rfsin, r = distance from pivot to force, F is the applied force
More informationCenter of Gravity Pearson Education, Inc.
Center of Gravity = The center of gravity position is at a place where the torque from one end of the object is balanced by the torque of the other end and therefore there is NO rotation. Fulcrum Point
More informationAP Physics QUIZ Chapters 10
Name: 1. Torque is the rotational analogue of (A) Kinetic Energy (B) Linear Momentum (C) Acceleration (D) Force (E) Mass A 5-kilogram sphere is connected to a 10-kilogram sphere by a rigid rod of negligible
More informationJordan University of Science & Technology PHYS 101A Final exam First semester 2007
Student Name Student ID Jordan University of Science & Technology PHYS 101A Final exam First semester 2007 Approximate your answer to those given for each question. Use this table to fill in your answer
More informationStructure of Biological Materials
ELEC ENG 3BA3: Structure of Biological Materials Notes for Lecture #7 Monday, September 24, 2012 3.2 Muscle biomechanics Organization: skeletal muscle is made up of muscle fibers each fiber is a single
More informationA. Incorrect! It looks like you forgot to include π in your calculation of angular velocity.
High School Physics - Problem Drill 10: Rotational Motion and Equilbrium 1. If a bike wheel of radius 50 cm rotates at 300 rpm what is its angular velocity and what is the linear speed of a point on the
More informationFind the component form of with initial point A(1, 3) and terminal point B(1, 3). Component form = 1 1, 3 ( 3) (x 1., y 1. ) = (1, 3) = 0, 6 Subtract.
Express a Vector in Component Form Find the component form of with initial point A(1, 3) and terminal point B(1, 3). = x 2 x 1, y 2 y 1 Component form = 1 1, 3 ( 3) (x 1, y 1 ) = (1, 3) and ( x 2, y 2
More informationCalculus with Analytic Geometry I Exam 10, Take Home Friday, November 8, 2013 Solutions.
All exercises are from Section 4.7 of the textbook. 1. Calculus with Analytic Geometry I Exam 10, Take Home Friday, November 8, 2013 Solutions. 2. Solution. The picture suggests using the angle θ as variable;
More informationDate Period Name. Energy, Work, and Simple Machines Vocabulary Review
Date Period Name CHAPTER 10 Study Guide Energy, Work, and Simple Machines Vocabulary Review Write the term that correctly completes the statement. Use each term once. compound machine joule resistance
More informationChapter 15+ Revisit Oscillations and Simple Harmonic Motion
Chapter 15+ Revisit Oscillations and Simple Harmonic Motion Revisit: Oscillations Simple harmonic motion To-Do: Pendulum oscillations Derive the parallel axis theorem for moments of inertia and apply it
More informationChapter 9. Rotational Dynamics
Chapter 9 Rotational Dynamics In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination of translation and rotation. 1) Torque Produces angular
More informationWORK, ENERGY, AND MACHINES
WORK, ENERGY, AND MACHINES Vocabulary Review Write the term that correctly completes the statement. Use each term once. compound machine joule resistance force efficiency kinetic energy translational kinetic
More informationGyroscopes and statics
Gyroscopes and statics Announcements: Welcome back from Spring Break! CAPA due Friday at 10pm We will finish Chapter 11 in H+R on angular momentum and start Chapter 12 on stability. Friday we will begin
More informationPhysics 5A Final Review Solutions
Physics A Final Review Solutions Eric Reichwein Department of Physics University of California, Santa Cruz November 6, 0. A stone is dropped into the water from a tower 44.m above the ground. Another stone
More informationBME 207 Introduction to Biomechanics Spring 2017
April 7, 2017 UNIVERSITY OF RHODE ISAND Department of Electrical, Computer and Biomedical Engineering BE 207 Introduction to Biomechanics Spring 2017 Homework 7 Problem 14.3 in the textbook. In addition
More informationChapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc.
Chapter 10 Rotational Kinematics and Energy 10-1 Angular Position, Velocity, and Acceleration 10-1 Angular Position, Velocity, and Acceleration Degrees and revolutions: 10-1 Angular Position, Velocity,
More informationName Student ID Score Last First. I = 2mR 2 /5 around the sphere s center of mass?
NOTE: ignore air resistance in all Questions. In all Questions choose the answer that is the closest!! Question I. (15 pts) Rotation 1. (5 pts) A bowling ball that has an 11 cm radius and a 7.2 kg mass
More informationName Date: Course number: MAKE SURE TA & TI STAMPS EVERY PAGE BEFORE YOU START. Grade: EXPERIMENT 4
Laboratory Section: Last Revised on June 18, 2018 Partners Names: Grade: EXPERIMENT 4 Moment of Inertia & Oscillations 0 Pre-Laboratory Work [20 pts] 1 a) In Section 31, describe briefly the steps you
More informationPHYSICS LAB Experiment 4 Fall 2004 ATWOOD S MACHINE: NEWTON S SECOND LAW
PHYSICS 83 - LAB Experiment 4 Fall 004 ATWOOD S MACHINE: NEWTON S SECOND LAW th In this experiment we will use a machine, used by George Atwood in the 8 century, to measure the gravitational acceleration,
More informationMotion in Space. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Motion in Space
Motion in Space MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Background Suppose the position vector of a moving object is given by r(t) = f (t), g(t), h(t), Background
More information1. The first thing you need to find is the mass of piece three. In order to find it you need to realize that the masses of the three pieces must be
1. The first thing you need to find is the mass of piece three. In order to find it you need to realize that the masses of the three pieces must be equal to the initial mass of the starting rocket. Now
More informationτ = F d Angular Kinetics Components of Torque (review from Systems FBD lecture Muscles Create Torques Torque is a Vector Work versus Torque
Components of Torque (review from Systems FBD lecture Angular Kinetics Hamill & Knutzen (Ch 11) Hay (Ch. 6), Hay & Ried (Ch. 12), Kreighbaum & Barthels (Module I & J) or Hall (Ch. 13 & 14) axis of rotation
More informationSimulation of the ergometer
Simulation of the ergometer 1. Introduction Why try to simulate the rowing ergometer on the computer? Through the performance monitor on the erg the user receives immediate feedback that he lacks in the
More informationAP Calculus Chapter 3 Testbank (Mr. Surowski)
AP Calculus Chapter 3 Testbank (Mr. Surowski) Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.). If f(x) = 0x 4 3 + x, then f (8) = (A) (B) 4 3 (C) 83 3 (D) 2 3 (E) 2
More informationChapter 12. Rotation of a Rigid Body
Chapter 12. Rotation of a Rigid Body Not all motion can be described as that of a particle. Rotation requires the idea of an extended object. This diver is moving toward the water along a parabolic trajectory,
More informationPittsfield High School Summer Assignment Contract Intensive 9 / Honors 9 / Honors Physics
Pittsfield High School Summer Assignment Contract Intensive 9 / Honors 9 / Honors Physics Welcome to Physics! The study of physics takes us on a journey investigating matter, energy, and how they interact.
More informationHomework #19 (due Friday 5/6)
Homework #19 (due Friday 5/6) Physics ID number Group Letter One issue that people often have trouble with at this point is distinguishing between tangential acceleration and centripetal acceleration for
More information= v 0 x. / t = 1.75m / s 2.25s = 0.778m / s 2 nd law taking left as positive. net. F x ! F
Multiple choice Problem 1 A 5.-N bos sliding on a rough horizontal floor, and the only horizontal force acting on it is friction. You observe that at one instant the bos sliding to the right at 1.75 m/s
More informationChapter 12: Rotation of Rigid Bodies. Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics
Chapter 1: Rotation of Rigid Bodies Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics Translational vs Rotational / / 1/ m x v dx dt a dv dt F ma p mv KE mv Work Fd P Fv / / 1/ I
More informationSection 3.4 Writing the Equation of a Line
Chapter Linear Equations and Functions Section.4 Writing the Equation of a Line Writing Equations of Lines Critical to a thorough understanding of linear equations and functions is the ability to write
More information10-6 Angular Momentum and Its Conservation [with Concept Coach]
OpenStax-CNX module: m50810 1 10-6 Angular Momentum and Its Conservation [with Concept Coach] OpenStax Tutor Based on Angular Momentum and Its Conservation by OpenStax College This work is produced by
More informationFigure Two. Then the two vector equations of equilibrium are equivalent to three scalar equations:
2004- v 10/16 2. The resultant external torque (the vector sum of all external torques) acting on the body must be zero about any origin. These conditions can be written as equations: F = 0 = 0 where the
More informationPositioning Servo Design Example
Positioning Servo Design Example 1 Goal. The goal in this design example is to design a control system that will be used in a pick-and-place robot to move the link of a robot between two positions. Usually
More informationPHYSICS - CLUTCH 1E CH 12: TORQUE & ROTATIONAL DYNAMICS.
!! www.clutchprep.com INTRO TO TORQUE TORQUE is a twist that a Force gives an object around an axis of rotation. - For example, when you push on a door, it rotates around its hinges. - When a Force acts
More informationPHYSICS - CLUTCH CH 12: TORQUE & ROTATIONAL DYNAMICS.
!! www.clutchprep.com TORQUE & ACCELERATION (ROTATIONAL DYNAMICS) When a Force causes rotation, it produces a Torque. Think of TORQUE as the equivalent of FORCE! FORCE (F) TORQUE (τ) - Causes linear acceleration
More informationChapter 2 Polynomial and Rational Functions
SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear
More informationGUIDED NOTES 2.2 LINEAR EQUATIONS IN ONE VARIABLE
GUIDED NOTES 2.2 LINEAR EQUATIONS IN ONE VARIABLE LEARNING OBJECTIVES In this section, you will: Solve equations in one variable algebraically. Solve a rational equation. Find a linear equation. Given
More informationDirectional Derivatives in the Plane
Directional Derivatives in the Plane P. Sam Johnson April 10, 2017 P. Sam Johnson (NIT Karnataka) Directional Derivatives in the Plane April 10, 2017 1 / 30 Directional Derivatives in the Plane Let z =
More informationUnit 7: Oscillations
Text: Chapter 15 Unit 7: Oscillations NAME: Problems (p. 405-412) #1: 1, 7, 13, 17, 24, 26, 28, 32, 35 (simple harmonic motion, springs) #2: 45, 46, 49, 51, 75 (pendulums) Vocabulary: simple harmonic motion,
More informationExam 1--PHYS 151--Chapter 1
ame: Class: Date: Exam 1--PHYS 151--Chapter 1 True/False Indicate whether the statement is true or false. Select A for True and B for False. 1. The force is a measure of an object s inertia. 2. Newton
More informationRegression. Estimation of the linear function (straight line) describing the linear component of the joint relationship between two variables X and Y.
Regression Bivariate i linear regression: Estimation of the linear function (straight line) describing the linear component of the joint relationship between two variables and. Generally describe as a
More informationINTRODUCTION TO DIFFERENTIATION
INTRODUCTION TO DIFFERENTIATION GRADIENT OF A CURVE We have looked at the process needed for finding the gradient of a curve (or the rate of change of a curve). We have defined the gradient of a curve
More informationWork - kinetic energy theorem for rotational motion *
OpenStax-CNX module: m14307 1 Work - kinetic energy theorem for rotational motion * Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0
More informationMODELING SINGLE LINK MOTION AND MUSCLE BEHAVIOR WITH A MODIFIED PENDULUM EQUATION
MODELING SINGLE LINK MOTION AND MUSCLE BEHAVIOR WITH A MODIFIED PENDULUM EQUATION By ALLISON SUTHERLIN A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON
More informationUncertainty, Error, and Precision in Quantitative Measurements an Introduction 4.4 cm Experimental error
Uncertainty, Error, and Precision in Quantitative Measurements an Introduction Much of the work in any chemistry laboratory involves the measurement of numerical quantities. A quantitative measurement
More informationBIOSTATISTICS NURS 3324
Simple Linear Regression and Correlation Introduction Previously, our attention has been focused on one variable which we designated by x. Frequently, it is desirable to learn something about the relationship
More informationFALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Saturday, 14 December 2013, 1PM to 4 PM, AT 1003
FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Saturday, 14 December 2013, 1PM to 4 PM, AT 1003 NAME: STUDENT ID: INSTRUCTION 1. This exam booklet has 14 pages. Make sure none are missing 2. There is
More informationChapter 10 Practice Test
Chapter 10 Practice Test 1. At t = 0, a wheel rotating about a fixed axis at a constant angular acceleration of 0.40 rad/s 2 has an angular velocity of 1.5 rad/s and an angular position of 2.3 rad. What
More informationBasic Physics. Isaac Newton ( ) Topics. Newton s Laws of Motion (2) Newton s Laws of Motion (1) PHYS 1411 Introduction to Astronomy
PHYS 1411 Introduction to Astronomy Basic Physics Chapter 5 Topics Newton s Laws Mass and Weight Work, Energy and Conservation of Energy Rotation, Angular velocity and acceleration Centripetal Force Angular
More information4 The Cartesian Coordinate System- Pictures of Equations
4 The Cartesian Coordinate System- Pictures of Equations Concepts: The Cartesian Coordinate System Graphs of Equations in Two Variables x-intercepts and y-intercepts Distance in Two Dimensions and the
More informationPhysics 201. Professor P. Q. Hung. 311B, Physics Building. Physics 201 p. 1/1
Physics 201 p. 1/1 Physics 201 Professor P. Q. Hung 311B, Physics Building Physics 201 p. 2/1 Rotational Kinematics and Energy Rotational Kinetic Energy, Moment of Inertia All elements inside the rigid
More informationKinesiology 201 Solutions Fluid and Sports Biomechanics
Kinesiology 201 Solutions Fluid and Sports Biomechanics Tony Leyland School of Kinesiology Simon Fraser University Fluid Biomechanics 1. Lift force is a force due to fluid flow around a body that acts
More informationRotation. I. Kinematics - Angular analogs
Rotation I. Kinematics - Angular analogs II. III. IV. Dynamics - Torque and Rotational Inertia Work and Energy Angular Momentum - Bodies and particles V. Elliptical Orbits The student will be able to:
More informationCIRCULAR MOTION AND ROTATION
1. UNIFORM CIRCULAR MOTION So far we have learned a great deal about linear motion. This section addresses rotational motion. The simplest kind of rotational motion is an object moving in a perfect circle
More informationSimple Linear Regression
9-1 l Chapter 9 l Simple Linear Regression 9.1 Simple Linear Regression 9.2 Scatter Diagram 9.3 Graphical Method for Determining Regression 9.4 Least Square Method 9.5 Correlation Coefficient and Coefficient
More informationBiostatistics and Design of Experiments Prof. Mukesh Doble Department of Biotechnology Indian Institute of Technology, Madras
Biostatistics and Design of Experiments Prof. Mukesh Doble Department of Biotechnology Indian Institute of Technology, Madras Lecture - 39 Regression Analysis Hello and welcome to the course on Biostatistics
More information