Aim : To study how the time period of a simple pendulum changes when its amplitude is changed.


 Charlotte Burke
 5 years ago
 Views:
Transcription
1 Aim : To study how the time period of a simple pendulum changes when its amplitude is changed. Teacher s Signature Name: Suvrat Raju Class: XIID Board Roll No.:
2 Table of Contents Aim Apparatus Theory Introduction General Discussion The time period for small amplitudes The equation of motion for large amplitudes The Jacobian elliptic functions Time period for large amplitudes Details about the method of experimentation Procedure Observations Calculations Graph Result Precautions Sources of Error Discussion Bibliography
3 Aim To study how the time period of a simple pendulum changes when its amplitude is changed. Suvrat Raju Physics Project 1
4 Apparatus Small heavy bob, String, Clamp Stand, Split Cork, Glass rod, Stop watch, Vernier Calipers, Scissors, Match Box, Candle. Suvrat Raju Physics Project 2
5 Theory Introduction General Discussion A simple pendulum ideally consists of a small heavy bob attached to a rigid support by means of a light inextensible string. When we speak of the motion of the simple pendulum, we refer to the oscillations it performs, when the bob is taken to a height (the string remaining taut) and released. Further, with reference to the above kind of motion, we define amplitude, timeperiod, and frequency as follows Amplitude : The amplitude of a simple pendulum is defined as the maximum angular deviation from the mean position of the bob. Oscillations : If the pendulum moves from one extreme position to the other and back to the first it is said to have performed one oscillation. Time Period : The time period of the simple pendulum is defined as the time required by the pendulum to complete one oscillation. Frequency : The frequency of the simple pendulum is defined as the number of oscillations performed per unit time. The time period for small amplitudes l θ For very small amplitudes, the motion of the simple pendulum may be approximated by simple harmonic motion. In this case we have : F = mg sin θ, (1) mg sinθ mg mg cosθ Fig. 1: The geometry of the simple pendulum where F is the restoring force acting on the pendulum, m is the mass of the bob, g is the acceleration due to gravity and θ is the angular displacement. Further, for small θ, Suvrat Raju Physics Project 3
6 sin θ θ. (2) Therefore, equation (1) reduces to F mg θ. (3) By Newton s second law of motion, F = mẋ.. (4) Here, x = l sin θ = l θ, by virtue of (2), so ẋ. = l θ.., and using (3) and (4), we get.. g θ = l θ (5) This is the same as the equation of motion for the simple harmonic motion θ. = p 2 θ, (6) with p 2 = g l, (7) which has the solution θ = A sin(pt+ϕ), (8) Suvrat Raju Physics Project 4
7 representing a simple harmonic motion with amplitude A, and initial phase ϕ. The time period for this simple harmonic motion is 2π. Putting back the value p of p from equation (7) we get the wellknown expression for the time period of the simple pendulum as: T = 2π l g (9) The equation of motion for large amplitudes The above expression, however, is not valid for large amplitudes, since the assumption sin θ θ no longer provides a good approximation for large θ. Therefore, we shall now endeavour to derive an exact equation of motion for the simple pendulum which will include the case of large amplitudes. We will see that the differential equation obtained is not analytically solvable. The analysis is interesting because it shows that the time period of the simple pendulum is dependent on the amplitude. Let l be the length of the pendulum and let P, K, and E be respectively the potential energy, kinetic energy and total energy of the system. Then P = mgl cos θ, (10) and K = 1 2 ml2 θ. 2 (11) since the velocity of the center of mass, at any instant, is lθ.. Hence, E = P + K = 1 2 ml2 θ. 2 mgl cos θ (12) Suvrat Raju Physics Project 5
8 At its maximum displacement the pendulum is instantaneously at rest. That is, if α is the amplitude of the pendulum, then for θ = α we have θ. = 0. Substituting in (12) we get E = mgl cos α (13) Therefore, equation (12) reduces to mgl cos α = 1 2 ml2 θ. 2 mgl cos θ (14) 1 2 ml2 θ. 2 = mgl cos θ mgl cos α θ. 2 = 2p 2 ( cos θ cos α ) = 4p 2 ( sin 2 α 2 sin2 θ 2 ), (15) where p 2 = g, as before, and the second equality follows from the wellknown identity cos θ = cos ( θ 2 + θ 2 ) = cos2 θ l 2 sin2 θ 2 = 1 2 sin2 θ 2. Let us define ϕ by sin ϕ = sin θ 2 sin α 2 sin θ 2 = sin α 2 sin ϕ. (16) Differentiating (16), we get Suvrat Raju Physics Project 6
9 1 2 cos θ 2 θ. = sin α 2 cos ϕ ϕ. (17) Multiplying (15) by 1 4 cos2 θ, we get cos2 θ 2 θ. 2 = p 2 cos 2 θ 2 ( sin2 α 2 sin2 θ 2 ). (18) The l.h.s. of (18) is the square of the l.h.s. of (17), so it may be replaced by the square of the r.h.s. of (17) to obtain sin 2 α 2 cos2 ϕ ϕ. 2 = p 2 cos 2 θ 2 ( sin2 α 2 sin2 θ 2 ). (19) Substituting for sin 2 θ 2 in the r.h.s. from (16), we obtain sin 2 α 2 cos2 ϕ ϕ. 2 = p 2 cos 2 θ 2 ( sin2 α 2 sin2 α 2 sin2 ϕ) (20) sin 2 α 2 cos2 ϕ ϕ. 2 = p 2 cos 2 θ 2 sin2 α 2 cos2 ϕ. (21) ϕ. 2 = p 2 (1 sin 2 θ 2 ), (22) upon canceling the common factors. Again using (16), ϕ. 2 = p 2 (1 sin 2 α 2 sin2 ϕ ). (23) Suvrat Raju Physics Project 7
10 If we multiply this equation by cos 2 ϕ, we get ϕ. 2 cos 2 ϕ = p 2 cos 2 ϕ(1 sin 2 α 2 sin2 ϕ). (24) If we now put, sin θ 2 y = sin ϕ = sin α, 2 (25) we have ẏ = ϕ. cos ϕ (26) and 1 y 2 = cos 2 ϕ. (27) Further, putting k = sin α 2 (28) and substituting in (24), we get ẏ 2 = p 2 (1 y 2 )(1 k 2 y 2 ). (29) Suvrat Raju Physics Project 8
11 To get rid of the constant p 2 let us define a new variable x = pt (30) so that dy dx = dy dt dx dt = ẏ p. (31) Substituting in (29) we get 2 dy dx = (1 y 2 )(1 k 2 y 2 ) (32) The Jacobian elliptic functions The above equation (32) cannot be solved in terms of elementary functions. Nor is the above form suited for numerical computations. Thus, it is obvious from physical considerations that the angular displacement of the pendulum reaches a maximum and then starts diminishing. Hence, from the definition of y in (25), and the definition of ϕ in (16), it is clear that y must reach a maximum and then start decreasing. Hence, the sign of dy will change as y dx crosses its maximum. But, for numerical computation, the sign of the square root must be specified unambiguously. In order to do this, we now need to transform the above equation. We define the function sn (x) to be that solution of (32) which satisfies the conditions sn(0) =0, sn (0) > 0. (33) Suvrat Raju Physics Project 9
12 Using sn (x), we define two more function cn (x) and dn (x) as follows cn 2 x = 1 sn 2 x, cn 0 = 1, (34) dn 2 x = 1 k 2 sn 2 x dn 0 = 1, (35) with the further condition that the function and their derivatives be continuous. Since sn (x) has been defined as a solution of equation (32), it satisfies that equation. That is, 2 d dx sn x = (1 sn 2 x)(1 k 2 sn 2 x). (36) Substituting from (34) and (35) we get 2 d dx sn x = cn 2 x dn 2 x. (37) We can now specify the sign of the square root in (32). This specification, due to the mathematician Jacobi, now takes on the elegant form d dx sn x = cn x dn x. (38) Differentiating (34), we get 2 cn x d dx cn x = 2 sn x d dx sn x = 2sn x cn x dn x. (39) Suvrat Raju Physics Project 10
13 where the second equality follows from an application of (38). Thus, canceling 2cn x from both sides, we obtain d dx cn x = sn x dn x (40) Similarly, differentiating (35), we get 2dn x d dx dn x = 2k2 sn x d dx sn x = 2k2 sn x cn x dn x where the second equality again follows from (38). The above equation simplifies to d dx dn x = k2 sn x cn x (41) These three simultaneous differential equations (38), (40), and (41) can now be numerically solved to obtain the values of sn x, cn x, dn x. This also solves the equation (32), which, by definition of sn x, has the solution y = sn x + constant. (42) Using the definition of y in (25), and the definition of x in (30), this allows us to express the angular displacement θ as a function of time t. θ = 2 sin 1 [sin α 2 sn(pt) + constant] (43) Suvrat Raju Physics Project 11
14 Time period for large amplitudes where the constant is fixed by the initial conditions. The three functions sn x, cn x, and dn x are known as the Jacobian elliptic functions. As can be seen from the definition (35), for k = 0, the function dn x 1. For this case of k = 0, the equations (38) and (34) are satisfied by the usual trigonometric functions, so that sn x reduces to sin x, and cn x reduces to cos x. Thus, the Jacobian elliptic functions may be considered generalizations of the trigonometric functions. The graphs of these periodic functions also are very similar to those of the trigonometric functions, though not identical. The results of numerically computing the timeperiod of sn x are given below. This computation shows that theoretically the time period of the simple pendulum must change with the amplitude From equation (43) we can see, that the θ is periodic with the same period as that of sn. Also, according to the standard result, sn x has the period 4K, where 4K = 0 1 dy (1 y 2 )(1 k 2 y 2) (44) Using x = pt, the time period of oscillation of the simple pendulum is given by T = 4K p = 4 p 0 1 dy (1 y 2 )(1 k 2 y 2) (45) substituting y = sin ϕ back into the equation, we get J.L Synge and B.A. Griffith, Principles of Mechanics, McGrawHill, New York, 1959, Third Edition, p 334. Suvrat Raju Physics Project 12
15 T = 4 p 0 π 2 dϕ (1 k 2 sin 2 ϕ) (46) As already observed, this elliptic integral must be evaluated numerically. But an analytical approximation may be used as a quick check of the numerical computation. On binomial expansion of the integrand in (46) we get, T = 4 p 0 π k 2 sin 2 ϕ k 4 sin 4 ϕ + + dϕ. (47) Now, 0 π 2 d ϕ = π 2, (48) while, I = sin 2 ϕ dϕ = sin ϕ sin ϕ = ( cos ϕ) sin ϕ dϕ = cos ϕ sin ϕ + cos 2 ϕ = cos ϕ sin ϕ + (1 sin 2 ϕ)dϕ = cos ϕ sin ϕ + dϕ sin 2 ϕ dϕ. Hence, 2I = cos ϕ sin ϕ + dϕ, which implies sin 2 ϕ = 1 2 dϕ 1 2 sin ϕ cos ϕ (49) so that, Suvrat Raju Physics Project 13
16 0 π 2 sin 2 ϕ d ϕ = 1 2 π 2. (50) Similarly, I 1 = sin 4 ϕ dϕ = sin 3 ϕ sin ϕ dϕ = sin 3 ϕ ( cos ϕ) dϕ = sin 3 ϕ cos ϕ + cos ϕ 3sin 2 ϕ cos ϕ dϕ = sin 3 ϕ cos ϕ + 3sin 2 ϕ(1 sin 2 ϕ)dϕ = sin 3 ϕ cos ϕ +3 sin 2 ϕ dϕ 3I 1, which implies sin 4 ϕ = 3 4 sin 2 ϕ dϕ 1 4 sin3 ϕ cos ϕ (51) so that 0 π 2 sin 4 ϕ d ϕ = π 2 (52) which gives us T = 2π p k k (53) Putting back p 2 = g and k = sin 1 α and taking only the first term in (53) we l 2 get as the first approximation, Suvrat Raju Physics Project 14
17 T = 2π l g (54) which agrees with equation (9). On taking the second term in (53) we get an improved second approximation, T = 2π l g 1 + α2 16 (55) which clearly shows us that the time period varies rather significantly with the amplitude. Details about the method of experimentation While trying to test the above theory, the primary practical problem that arose was to find a method to measure the amplitude accurately. The second problem was to ensure that the bob oscillated in one plane. The third problem was to ensure, that no force was inadvertently applied while releasing the bob. All three problems were solved by using the apparatus detailed in the diagram. A θ String(to be cut) String C Bob Clamp Stand B Mainly, the apparatus consisted of a clamp stand to which a split cork was attached, and through which the string BC(see figure) was passed and tied to the bob. By varying the length of the string, the length of the simple pendulum could be varied. Further, a horizontal glass rod was also attached to the stand. Now another string of known length AC was tied to the bob, and was looped over the glass rod as shown. By measuring AB and using the two known lengths AC and BC, the angle θ could be calculated using trigonometry. Fig: 2 A shematic depiction of the appratus Now, when the reading was to be taken, the string attached to the glass rod was cut causing the bob to start oscillating. This ensured that no external force was applied to the bob, and this also ensured that it moved in a single plane. Suvrat Raju Physics Project 15
18 Procedure 1) First the radius of the bob was measured using Vernier calipers. 2) Now a string was tied to the bob, and a length of 100 cm(including the hook and the radius of the bob) was marked. 3) The string was now passed through a split cork attached to a clamp stand. It was lowered, till the length through the split cork exactly equaled 100 cm. 4) Now, a horizontal glass rod was also clamped to the stand at one end and another stand, exactly parallel to the original stand, at the other end. The arrangement was such that the top of the rod touched the split cork. 5) Now another string was tied to the bob. A known length was marked out on it. 6) This string was now tied at the glass rod, at a specific distance from the split cork so that the amplitude equaled a desired angle. 7) Now this second string was burnt, using a candle, and the stop watch was started. 8) After a specified number of oscillations, the stop watch was stopped. The elapsed time was divided by the number of oscillations to give the time period. 9) This procedure was repeated both for different angles and for different lengths of the pendulum. Suvrat Raju Physics Project 16
19 Observations Two sets of observations were taken. In one, the length of the simple pendulum was 100 cm and 20 oscillations were timed. In the other, the length of the pendulum was 80 cm and 10 oscillations were timed. These numbers were taken, because for a larger number of oscillations, the amplitude was observed to change considerably due to damping. This was also done, in order to get maximum variety, as factors such as damping would effect the second set less, but random errors would be minimized in the first. Least count of the stop watch 0.1 s Least count of the metre scale 0.1 cm radius of the bob 0.9 cm Length of the pendulum 100 cm No of oscillations timed 20 S. No Amplitude ( 0 ) Time required (s) Observed Time period (s) Expected Time period (s) Classical Time period (s) Suvrat Raju Physics Project 17
20 Length of the pendulum 80 cm No of oscillations timed 10 S. No Amplitude( 0 ) Time required (s) Observed Time Period (s) Expected Time Period (s) Classical Time period (s) Suvrat Raju Physics Project 18
21 Calculations The values in the expected time period column of the observation table were obtained by calculating the time period by simultaneously solving the system of equations (38), (40) and (41) numerically. All computations for numerical solution of the differential equations were carried out on a 486 DX2 processor running DOS 6.22 using a double precision 8th order Runge Kutta algorithm (the coefficients of Dormand and Prince) in the executable software CALCODE. The software also computes the sections of the solution using a highorder interpolation scheme, and the regula falsi method. The time period was calculated by noting the difference between successive zeros of the function sn(x). Inputs to the executable are the three firstorder equations (38), (40), (41), entered symbolically, with the convention that y 1 = sn, y 2 = cn y 3 = dn (56) The input parameters for these equations are the values of p, and k given respectively by (7), and (28): these values can evidently be calculated from a knowledge of the length of the string, l, the acceleration due to gravity, g, and the amplitude α, and are tabulated below for the actual settings used. These values of p, and k suffice to fix the time period of oscillation. A small auxiliary computer program was written to calculate p and k. The initial data are the same in all cases, and are given by (33), (34), and (35), viz., y 1 (0) = 0, y 2 (0) = 1, y 3 (0) = 1. The table below also compares the time period calculated numerically by the above sophisticated numerical method with the approximation given by (55). Suvrat Raju Physics Project 19
22 p = (g l) 1 2 (l = length) S. No. α k = sin α (α = amplitude) (l = 1.00 m) (l = 0.8 m) Time period from CALCODE 2π p Time period from equation (55) Suvrat Raju Physics Project 20
23 Graph Two graphs were plotted, one for each set of readings. The classical time period, the expected time period, and the observed time period were depicted on the graph together. Suvrat Raju Physics Project 21
24 Result The variation in the time period of a simple pendulum, on change in amplitude was studied. (i) An accurate expression for derived both for the equation of motion and the time period for large amplitudes. This was verified within experimental errors. (ii) A graph was plotted between the Amplitude and the Time Period. Suvrat Raju Physics Project 22
25 Precautions 1) The length of the simple pendulum must take into account, both the length of the hook and the radius of the bob. 2)The string should not be stretched excessively, while measuring the length. 3)The stop watch should be started immediately after the second supporting string is burnt. 4)The glass rod, should be completely horizontal. 5)The glass rod, should touch the cork, and should not have a very large diameter, as it is assumed to be a straight line, in calculations. 6)Before the string is burnt, the bob should be absolutely stationary. During motion, care should be taken, that no torque acts on the bob, and the string does not coil up on itself. Suvrat Raju Physics Project 23
26 Sources of Error 1) The major source of error was damping due to friction at the point of suspension, and also due to air. Due to this friction, the total energy of the bob does not remain exactly constant. Instead, the total energy continuously decreases along with the amplitude. Hence, also, the time period changes from oscillation to oscillation. The experimental arrangement, therefore, does not reflect the theoretical assumptions very accurately. Hence one can observe a systematic departure from the theoretically calculated values. 2)Extraneous sources, such as air currents can also cause random errors. 3) Errors are also caused by the least count of the stop watch and the metre scale, the thickness of the glass rod, the slight extensibility and mass of the string and the slight non spherical shape of the bob Suvrat Raju Physics Project 24
27 Discussion The graph of the Time Period vs the Amplitude clearly shows that though the theory provides a very close approximation to the observations initially, a steady deviation is shown as the amplitude increases. This is partly because of the various sources of error like the extensibility of the string, the slight nonspherical shape of the bob, the least count of the stop watch etc. but also partly because the modeling of the physical situation is not exact. The project may be improved by using a more sophisticated theoretical model which takes damping into account. This would also require a superior apparatus. A better quality string, a thinner glass rod, and a stop watch with a smaller least count would be useful. A heavier bob would allow more oscillations to be timed, and would improve accuracy. Suvrat Raju Physics Project 25
28 Bibliography J.L Synge and B.A Griffith, Principles of Mechanics, McGrawHill, New York, 1959, pp G. F Simmons, Differential Equations, Tata McGrawHill, New Delhi, 1972, pp D. Halliday, R. Resnick and K.S Krane, Physics, Wiley, Singapore, 1994, pp Suvrat Raju Physics Project 26
Determination of Acceleration due to Gravity
Experiment 2 24 Kuwait University Physics 105 Physics Department Determination of Acceleration due to Gravity Introduction In this experiment the acceleration due to gravity (g) is determined using two
More informationPENDULUM PERIODS. First Last. Partners: student1, student2, and student3
PENDULUM PERIODS First Last Partners: student1, student2, and student3 Governor s School for Science and Technology 520 Butler Farm Road, Hampton, VA 23666 April 13, 2011 ABSTRACT The effect of amplitude,
More informationSpring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations
Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring
More informationSimple Harmonic Motion
Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights
More informationUnit  6 Vibrations of Two Degree of Freedom Systems
Unit  6 Vibrations of Two Degree of Freedom Systems Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Introduction A two
More informationAP1 Oscillations. 1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More informationExperiment 9. The Pendulum
Experiment 9 The Pendulum 9.1 Objectives Investigate the functional dependence of the period (τ) 1 of a pendulum on its length (L), the mass of its bob (m), and the starting angle (θ 0 ). Use a pendulum
More informationPhysics 41 HW Set 1 Chapter 15
Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,
More informationwww.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x
Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity
More informationPrelab Exercises: Hooke's Law and the Behavior of Springs
59 Prelab Exercises: Hooke's Law and the Behavior of Springs Study the description of the experiment that follows and answer the following questions.. (3 marks) Explain why a mass suspended vertically
More informationAcceleration due to Gravity
Acceleration due to Gravity 1 Object To determine the acceleration due to gravity by different methods. 2 Apparatus Balance, ball bearing, clamps, electric timers, meter stick, paper strips, precision
More informationLab 7: Rotational Motion
Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME9472), string with loop at one end and small white bead at the other end (125
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationAP Physics C. Oscillations/SHM Review Packet
AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete
More informationLecture L222D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L  D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L3 for
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationApplications of SecondOrder Differential Equations
Applications of SecondOrder Differential Equations Secondorder linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration
More informationSOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
More information2 Session Two  Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision  Session 2: Complex Numbers and Vectors 1 2 Session Two  Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More informationA Determination of g, the Acceleration Due to Gravity, from Newton's Laws of Motion
A Determination of g, the Acceleration Due to Gravity, from Newton's Laws of Motion Objective In the experiment you will determine the cart acceleration, a, and the friction force, f, experimentally for
More informationMidterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m
Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of
More informationHOOKE S LAW AND SIMPLE HARMONIC MOTION
HOOKE S LAW AND SIMPLE HARMONIC MOTION Alexander Sapozhnikov, Brooklyn College CUNY, New York, alexs@brooklyn.cuny.edu Objectives Study Hooke s Law and measure the spring constant. Study Simple Harmonic
More informationD Alembert s principle and applications
Chapter 1 D Alembert s principle and applications 1.1 D Alembert s principle The principle of virtual work states that the sum of the incremental virtual works done by all external forces F i acting in
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationLecture L5  Other Coordinate Systems
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5  Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com
Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations
More informationEDUMECH Mechatronic Instructional Systems. Ball on Beam System
EDUMECH Mechatronic Instructional Systems Ball on Beam System Product of Shandor Motion Systems Written by Robert Hirsch Ph.D. 9989 All Rights Reserved. 999 Shandor Motion Systems, Ball on Beam Instructional
More informationELASTIC FORCES and HOOKE S LAW
PHYS101 LAB03 ELASTIC FORCES and HOOKE S LAW 1. Objective The objective of this lab is to show that the response of a spring when an external agent changes its equilibrium length by x can be described
More informationIf you put the same book on a tilted surface the normal force will be less. The magnitude of the normal force will equal: N = W cos θ
Experiment 4 ormal and Frictional Forces Preparation Prepare for this week's quiz by reviewing last week's experiment Read this week's experiment and the section in your textbook dealing with normal forces
More informationHOOKE S LAW AND OSCILLATIONS
9 HOOKE S LAW AND OSCILLATIONS OBJECTIVE To measure the effect of amplitude, mass, and spring constant on the period of a springmass oscillator. INTRODUCTION The force which restores a spring to its equilibrium
More informationRotation: 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
More informationPendulum Force and Centripetal Acceleration
Pendulum Force and Centripetal Acceleration 1 Objectives 1. To calibrate and use a force probe and motion detector. 2. To understand centripetal acceleration. 3. To solve force problems involving centripetal
More informationDetermination of g using a spring
INTRODUCTION UNIVERSITY OF SURREY DEPARTMENT OF PHYSICS Level 1 Laboratory: Introduction Experiment Determination of g using a spring This experiment is designed to get you confident in using the quantitative
More informationHIGH VOLTAGE ELECTROSTATIC PENDULUM
HIGH VOLTAGE ELECTROSTATIC PENDULUM Raju Baddi National Center for Radio Astrophysics, TIFR, Ganeshkhind P.O Bag 3, Pune University Campus, PUNE 411007, Maharashtra, INDIA; baddi@ncra.tifr.res.in ABSTRACT
More informationLecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is
Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.49.6, 10.110.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of
More informationPhysics 1120: Simple Harmonic Motion Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured
More informationRotational Inertia Demonstrator
WWW.ARBORSCI.COM Rotational Inertia Demonstrator P33545 BACKGROUND: The Rotational Inertia Demonstrator provides an engaging way to investigate many of the principles of angular motion and is intended
More informationPhysics 231 Lecture 15
Physics 31 ecture 15 Main points of today s lecture: Simple harmonic motion Mass and Spring Pendulum Circular motion T 1/f; f 1/ T; ω πf for mass and spring ω x Acos( ωt) v ωasin( ωt) x ax ω Acos( ωt)
More informationCentripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.
Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.
More informationPhysics 211: Lab Oscillations. Simple Harmonic Motion.
Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.
More informationPHYS 2425 Engineering Physics I EXPERIMENT 9 SIMPLE HARMONIC MOTION
PHYS 2425 Engineering Physics I EXPERIMENT 9 SIMPLE HARMONIC MOTION I. INTRODUCTION The objective of this experiment is the study of oscillatory motion. In particular the springmass system and the simple
More informationChapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.
Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems
More informationLecture L2  Degrees of Freedom and Constraints, Rectilinear Motion
S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L  Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates
More informationSOLID MECHANICS DYNAMICS TUTORIAL MOMENT OF INERTIA. This work covers elements of the following syllabi.
SOLID MECHANICS DYNAMICS TUTOIAL MOMENT OF INETIA This work covers elements of the following syllabi. Parts of the Engineering Council Graduate Diploma Exam D5 Dynamics of Mechanical Systems Parts of the
More informationC B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N
Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a
More informationTorque and Rotary Motion
Torque and Rotary Motion Name Partner Introduction Motion in a circle is a straightforward extension of linear motion. According to the textbook, all you have to do is replace displacement, velocity,
More informationChapter 18 Static Equilibrium
Chapter 8 Static Equilibrium 8. Introduction Static Equilibrium... 8. Lever Law... Example 8. Lever Law... 4 8.3 Generalized Lever Law... 5 8.4 Worked Examples... 7 Example 8. Suspended Rod... 7 Example
More information226 Chapter 15: OSCILLATIONS
Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion
More informationReflection and Refraction
Equipment Reflection and Refraction Acrylic block set, planeconcaveconvex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass
Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More informationFRICTION, WORK, AND THE INCLINED PLANE
FRICTION, WORK, AND THE INCLINED PLANE Objective: To measure the coefficient of static and inetic friction between a bloc and an inclined plane and to examine the relationship between the plane s angle
More informationA) F = k x B) F = k C) F = x k D) F = x + k E) None of these.
CT161 Which of the following is necessary to make an object oscillate? i. a stable equilibrium ii. little or no friction iii. a disturbance A: i only B: ii only C: iii only D: i and iii E: All three Answer:
More informationAP PHYSICS C Mechanics  SUMMER ASSIGNMENT FOR 20162017
AP PHYSICS C Mechanics  SUMMER ASSIGNMENT FOR 20162017 Dear Student: The AP physics course you have signed up for is designed to prepare you for a superior performance on the AP test. To complete material
More informationPractice Test SHM with Answers
Practice Test SHM with Answers MPC 1) If we double the frequency of a system undergoing simple harmonic motion, which of the following statements about that system are true? (There could be more than one
More informationChapter 6 Work and Energy
Chapter 6 WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system
More informationPhysics 1A Lecture 10C
Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. Oprah Winfrey Static Equilibrium
More informationboth double. A. T and v max B. T remains the same and v max doubles. both remain the same. C. T and v max
Q13.1 An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object s maximum speed
More informationWeight The weight of an object is defined as the gravitational force acting on the object. Unit: Newton (N)
Gravitational Field A gravitational field as a region in which an object experiences a force due to gravitational attraction Gravitational Field Strength The gravitational field strength at a point in
More informationSimple Harmonic Motion Experiment. 1 f
Simple Harmonic Motion Experiment In this experiment, a motion sensor is used to measure the position of an oscillating mass as a function of time. The frequency of oscillations will be obtained by measuring
More informationNewton s Second Law. ΣF = m a. (1) In this equation, ΣF is the sum of the forces acting on an object, m is the mass of
Newton s Second Law Objective The Newton s Second Law experiment provides the student a hands on demonstration of forces in motion. A formulated analysis of forces acting on a dynamics cart will be developed
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationReview D: Potential Energy and the Conservation of Mechanical Energy
MSSCHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.01 Fall 2005 Review D: Potential Energy and the Conservation of Mechanical Energy D.1 Conservative and Nonconservative Force... 2 D.1.1 Introduction...
More informationIntroduction to Complex Numbers in Physics/Engineering
Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The
More informationPractice Exam Three Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004 Practice Exam Three Solutions Problem 1a) (5 points) Collisions and Center of Mass Reference Frame In the lab frame,
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME  TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More informationTorque Analyses of a Sliding Ladder
Torque Analyses of a Sliding Ladder 1 Problem Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (May 6, 2007) The problem of a ladder that slides without friction while
More informationPHYS 211 FINAL FALL 2004 Form A
1. Two boys with masses of 40 kg and 60 kg are holding onto either end of a 10 m long massless pole which is initially at rest and floating in still water. They pull themselves along the pole toward each
More informationv v ax v a x a v a v = = = Since F = ma, it follows that a = F/m. The mass of the arrow is unchanged, and ( )
Week 3 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationSample lab procedure and report. The Simple Pendulum
Sample lab procedure and report The Simple Pendulum In this laboratory, you will investigate the effects of a few different physical variables on the period of a simple pendulum. The variables we consider
More informationSample Questions for the AP Physics 1 Exam
Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiplechoice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each
More informationAwellknown lecture demonstration1
Acceleration of a Pulled Spool Carl E. Mungan, Physics Department, U.S. Naval Academy, Annapolis, MD 40506; mungan@usna.edu Awellknown lecture demonstration consists of pulling a spool by the free end
More information2.2 Magic with complex exponentials
2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or
More informationCh 7 Kinetic Energy and Work. Question: 7 Problems: 3, 7, 11, 17, 23, 27, 35, 37, 41, 43
Ch 7 Kinetic Energy and Work Question: 7 Problems: 3, 7, 11, 17, 23, 27, 35, 37, 41, 43 Technical definition of energy a scalar quantity that is associated with that state of one or more objects The state
More informationPhysics Notes Class 11 CHAPTER 6 WORK, ENERGY AND POWER
1 P a g e Work Physics Notes Class 11 CHAPTER 6 WORK, ENERGY AND POWER When a force acts on an object and the object actually moves in the direction of force, then the work is said to be done by the force.
More informationDerive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 115 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
More informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationCambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level
Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level *0123456789* PHYSICS 9702/02 Paper 2 AS Level Structured Questions For Examination from 2016 SPECIMEN
More informationPHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013
PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be
More informationOscillations: Mass on a Spring and Pendulums
Chapter 3 Oscillations: Mass on a Spring and Pendulums 3.1 Purpose 3.2 Introduction Galileo is said to have been sitting in church watching the large chandelier swinging to and fro when he decided that
More informationPHY121 #8 Midterm I 3.06.2013
PHY11 #8 Midterm I 3.06.013 AP Physics Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension
More information3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH. In Isaac Newton's day, one of the biggest problems was poor navigation at sea.
BA01 ENGINEERING MATHEMATICS 01 CHAPTER 3 APPLICATION OF DIFFERENTIATION 3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH Introduction to Applications of Differentiation In Isaac Newton's
More informationPrecise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility
Precise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility Renuka V. S. & Abraham T Mathew Electrical Engineering Department, NIT Calicut Email : renuka_mee@nitc.ac.in,
More informationConservation of Energy Physics Lab VI
Conservation of Energy Physics Lab VI Objective This lab experiment explores the principle of energy conservation. You will analyze the final speed of an air track glider pulled along an air track by a
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More information19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonlyoccurring firstorder and secondorder ordinary differential equations.
More informationRotational Motion: Moment of Inertia
Experiment 8 Rotational Motion: Moment of Inertia 8.1 Objectives Familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid body
More informationSURFACE TENSION. Definition
SURFACE TENSION Definition In the fall a fisherman s boat is often surrounded by fallen leaves that are lying on the water. The boat floats, because it is partially immersed in the water and the resulting
More informationAS COMPETITION PAPER 2008
AS COMPETITION PAPER 28 Name School Town & County Total Mark/5 Time Allowed: One hour Attempt as many questions as you can. Write your answers on this question paper. Marks allocated for each question
More informationProblem Set 5 Work and Kinetic Energy Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physics Physics 8.1 Fall 1 Problem Set 5 Work and Kinetic Energy Solutions Problem 1: Work Done by Forces a) Two people push in opposite directions on
More informationHSC Mathematics  Extension 1. Workshop E4
HSC Mathematics  Extension 1 Workshop E4 Presented by Richard D. Kenderdine BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat School of Mathematics and Applied Statistics University of Wollongong
More informationReview A: Vector Analysis
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Review A: Vector Analysis A... A0 A.1 Vectors A2 A.1.1 Introduction A2 A.1.2 Properties of a Vector A2 A.1.3 Application of Vectors
More informationPhysics 40 Lab 1: Tests of Newton s Second Law
Physics 40 Lab 1: Tests of Newton s Second Law January 28 th, 2008, Section 2 Lynda Williams Lab Partners: Madonna, Hilary Clinton & Angie Jolie Abstract Our primary objective was to test the validity
More informationMechanics. Determining the gravitational constant with the gravitation torsion balance after Cavendish. LD Physics Leaflets P1.1.3.1.
Mechanics Measuring methods Determining the gravitational constant LD Physics Leaflets P1.1.3.1 Determining the gravitational constant with the gravitation torsion balance after Cavendish Measuring the
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationLecture L6  Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6  Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationENERGYand WORK (PART I and II) 9MAC
ENERGYand WORK (PART I and II) 9MAC Purpose: To understand work, potential energy, & kinetic energy. To understand conservation of energy and how energy is converted from one form to the other. Apparatus:
More information