/FontDescriptor 11 0 R 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 Compare it to the equation for a straight line. 12 0 obj In addition, there are hundreds of problems with detailed solutions on various physics topics. This paper presents approximate periodic solutions to the anharmonic (i.e. /FontDescriptor 11 0 R WebPENDULUM WORKSHEET 1. A simple pendulum is defined to have a point mass, also known as the pendulum bob, which is suspended from a string of length L with negligible mass (Figure 15.5.1 ). /BaseFont/VLJFRF+CMMI8 323.4 354.2 600.2 323.4 938.5 631 569.4 631 600.2 446.4 452.6 446.4 631 600.2 815.5 << 30 0 obj endobj /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 are licensed under a, Introduction: The Nature of Science and Physics, Introduction to Science and the Realm of Physics, Physical Quantities, and Units, Accuracy, Precision, and Significant Figures, Introduction to One-Dimensional Kinematics, Motion Equations for Constant Acceleration in One Dimension, Problem-Solving Basics for One-Dimensional Kinematics, Graphical Analysis of One-Dimensional Motion, Introduction to Two-Dimensional Kinematics, Kinematics in Two Dimensions: An Introduction, Vector Addition and Subtraction: Graphical Methods, Vector Addition and Subtraction: Analytical Methods, Dynamics: Force and Newton's Laws of Motion, Introduction to Dynamics: Newtons Laws of Motion, Newtons Second Law of Motion: Concept of a System, Newtons Third Law of Motion: Symmetry in Forces, Normal, Tension, and Other Examples of Forces, Further Applications of Newtons Laws of Motion, Extended Topic: The Four Basic ForcesAn Introduction, Further Applications of Newton's Laws: Friction, Drag, and Elasticity, Introduction: Further Applications of Newtons Laws, Introduction to Uniform Circular Motion and Gravitation, Fictitious Forces and Non-inertial Frames: The Coriolis Force, Satellites and Keplers Laws: An Argument for Simplicity, Introduction to Work, Energy, and Energy Resources, Kinetic Energy and the Work-Energy Theorem, Introduction to Linear Momentum and Collisions, Collisions of Point Masses in Two Dimensions, Applications of Statics, Including Problem-Solving Strategies, Introduction to Rotational Motion and Angular Momentum, Dynamics of Rotational Motion: Rotational Inertia, Rotational Kinetic Energy: Work and Energy Revisited, Collisions of Extended Bodies in Two Dimensions, Gyroscopic Effects: Vector Aspects of Angular Momentum, Variation of Pressure with Depth in a Fluid, Gauge Pressure, Absolute Pressure, and Pressure Measurement, Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action, Fluid Dynamics and Its Biological and Medical Applications, Introduction to Fluid Dynamics and Its Biological and Medical Applications, The Most General Applications of Bernoullis Equation, Viscosity and Laminar Flow; Poiseuilles Law, Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes, Temperature, Kinetic Theory, and the Gas Laws, Introduction to Temperature, Kinetic Theory, and the Gas Laws, Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature, Introduction to Heat and Heat Transfer Methods, The First Law of Thermodynamics and Some Simple Processes, Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency, Carnots Perfect Heat Engine: The Second Law of Thermodynamics Restated, Applications of Thermodynamics: Heat Pumps and Refrigerators, Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy, Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation, Introduction to Oscillatory Motion and Waves, Hookes Law: Stress and Strain Revisited, Simple Harmonic Motion: A Special Periodic Motion, Energy and the Simple Harmonic Oscillator, Uniform Circular Motion and Simple Harmonic Motion, Speed of Sound, Frequency, and Wavelength, Sound Interference and Resonance: Standing Waves in Air Columns, Introduction to Electric Charge and Electric Field, Static Electricity and Charge: Conservation of Charge, Electric Field: Concept of a Field Revisited, Conductors and Electric Fields in Static Equilibrium, Introduction to Electric Potential and Electric Energy, Electric Potential Energy: Potential Difference, Electric Potential in a Uniform Electric Field, Electrical Potential Due to a Point Charge, Electric Current, Resistance, and Ohm's Law, Introduction to Electric Current, Resistance, and Ohm's Law, Ohms Law: Resistance and Simple Circuits, Alternating Current versus Direct Current, Introduction to Circuits and DC Instruments, DC Circuits Containing Resistors and Capacitors, Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field, Force on a Moving Charge in a Magnetic Field: Examples and Applications, Magnetic Force on a Current-Carrying Conductor, Torque on a Current Loop: Motors and Meters, Magnetic Fields Produced by Currents: Amperes Law, Magnetic Force between Two Parallel Conductors, Electromagnetic Induction, AC Circuits, and Electrical Technologies, Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies, Faradays Law of Induction: Lenzs Law, Maxwells Equations: Electromagnetic Waves Predicted and Observed, Introduction to Vision and Optical Instruments, Limits of Resolution: The Rayleigh Criterion, *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light, Photon Energies and the Electromagnetic Spectrum, Probability: The Heisenberg Uncertainty Principle, Discovery of the Parts of the Atom: Electrons and Nuclei, Applications of Atomic Excitations and De-Excitations, The Wave Nature of Matter Causes Quantization, Patterns in Spectra Reveal More Quantization, Introduction to Radioactivity and Nuclear Physics, Introduction to Applications of Nuclear Physics, The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited, Particles, Patterns, and Conservation Laws, A simple pendulum has a small-diameter bob and a string that has a very small mass but is strong enough not to stretch appreciably. We see from Figure 16.13 that the net force on the bob is tangent to the arc and equals mgsinmgsin. 24/7 Live Expert. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 706.4 938.5 877 781.8 754 843.3 815.5 877 815.5 1. 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 The equation of frequency of the simple pendulum : f = frequency, g = acceleration due to gravity, l = the length of cord. This is the video that cover the section 7. The rst pendulum is attached to a xed point and can freely swing about it. /Name/F5 if(typeof ez_ad_units != 'undefined'){ez_ad_units.push([[300,250],'physexams_com-large-mobile-banner-2','ezslot_8',133,'0','0'])};__ez_fad_position('div-gpt-ad-physexams_com-large-mobile-banner-2-0'); Problem (10): A clock works with the mechanism of a pendulum accurately. 799.2 642.3 942 770.7 799.4 699.4 799.4 756.5 571 742.3 770.7 770.7 1056.2 770.7 /BaseFont/SNEJKL+CMBX12 Boundedness of solutions ; Spring problems . /FirstChar 33 /Type/Font WebAnalytic solution to the pendulum equation for a given initial conditions and Exact solution for the nonlinear pendulum (also here). Hence, the length must be nine times. << A 2.2 m long simple pendulum oscillates with a period of 4.8 s on the surface of Pendulum clocks really need to be designed for a location. What is the period of oscillations? @bL7]qwxuRVa1Z/. HFl`ZBmMY7JHaX?oHYCBb6#'\ }! Solution: (a) the number of complete cycles $N$ in a specific time interval $t$ is defined as the frequency $f$ of an oscillatory system or \[f=\frac{N}{t}\] Therefore, the frequency of this pendulum is calculated as \[f=\frac{50}{40\,{\rm s}}=1.25\, {\rm Hz}\] Solution: Recall that the time period of a clock pendulum, which is the time between successive ticks (one complete cycle), is proportional to the inverse of the square root of acceleration of gravity, $T\propto 1/\sqrt{g}$. Both are suspended from small wires secured to the ceiling of a room. @ @y ss~P_4qu+a" ' 9y c&Ls34f?q3[G)> `zQGOxis4t&0tC: pO+UP=ebLYl*'zte[m04743C 3d@C8"P)Dp|Y Let us define the potential energy as being zero when the pendulum is at the bottom of the swing, = 0 . 0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 We can discern one half the smallest division so DVVV= ()05 01 005.. .= VV V= D ()385 005.. 4. endobj /LastChar 196 Electric generator works on the scientific principle. endobj 12 0 obj Knowing 323.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 323.4 323.4 10 0 obj To verify the hypothesis that static coefficients of friction are dependent on roughness of surfaces, and independent of the weight of the top object. /FirstChar 33 9 0 obj Arc length and sector area worksheet (with answer key) Find the arc length. Find the period and oscillation of this setup. Look at the equation again. If displacement from equilibrium is very small, then the pendulum of length $\ell$ approximate simple harmonic motion. Or at high altitudes, the pendulum clock loses some time. << /Pages 45 0 R /Type /Catalog >> If, is the frequency of the first pendulum and, is the frequency of the second pendulum, then determine the relationship between, Based on the equation above, can conclude that, ased on the above formula, can conclude the length of the, (l) and the acceleration of gravity (g) impact the period of, determine the length of rope if the frequency is twice the initial frequency. >> 384.3 611.1 611.1 611.1 611.1 611.1 896.3 546.3 611.1 870.4 935.2 611.1 1077.8 1207.4 B ased on the above formula, can conclude the length of the rod (l) and the acceleration of gravity (g) impact the period of the simple pendulum. A7)mP@nJ WebWalking up and down a mountain. xcbd`g`b``8 "w ql6A$7d s"2Z RQ#"egMf`~$ O t y y=1 y=0 Fig. H then you must include on every digital page view the following attribution: Use the information below to generate a citation. can be very accurate. The digital stopwatch was started at a time t 0 = 0 and then was used to measure ten swings of a /Type/Font Note how close this is to one meter. /Font <>>> 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] For small displacements, a pendulum is a simple harmonic oscillator. << 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.3 856.5 799.4 713.6 685.2 770.7 742.3 799.4 The period of the Great Clock's pendulum is probably 4seconds instead of the crazy decimal number we just calculated. 805.5 896.3 870.4 935.2 870.4 935.2 0 0 870.4 736.1 703.7 703.7 1055.5 1055.5 351.8 The length of the second pendulum is 0.4 times the length of the first pendulum, and the acceleration of gravity experienced by the second pendulum is 0.9 times the acceleration of gravity experienced by the first pendulum. /LastChar 196 Problem (12): If the frequency of a 69-cm-long pendulum is 0.601 Hz, what is the value of the acceleration of gravity $g$ at that location? /FirstChar 33 Use the pendulum to find the value of gg on planet X. 511.1 511.1 511.1 831.3 460 536.7 715.6 715.6 511.1 882.8 985 766.7 255.6 511.1] 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 896.3 896.3 740.7 351.8 611.1 351.8 611.1 351.8 351.8 611.1 675.9 546.3 675.9 546.3 WebPeriod and Frequency of a Simple Pendulum: Class Work 27. 460 664.4 463.9 485.6 408.9 511.1 1022.2 511.1 511.1 511.1 0 0 0 0 0 0 0 0 0 0 0 % >> WebThe simple pendulum is another mechanical system that moves in an oscillatory motion. /LastChar 196 /FontDescriptor 20 0 R << If the length of the cord is increased by four times the initial length, then determine the period of the harmonic motion. stream /Subtype/Type1 /Widths[306.7 514.4 817.8 769.1 817.8 766.7 306.7 408.9 408.9 511.1 766.7 306.7 357.8 Figure 2: A simple pendulum attached to a support that is free to move. How accurate is this measurement? Free vibrations ; Damped vibrations ; Forced vibrations ; Resonance ; Nonlinear models ; Driven models ; Pendulum . not harmonic or non-sinusoidal) response of a simple pendulum undergoing moderate- to large-amplitude oscillations. endobj 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 The governing differential equation for a simple pendulum is nonlinear because of the term. WebPhysics 1 Lab Manual1Objectives: The main objective of this lab is to determine the acceleration due to gravity in the lab with a simple pendulum. xYK WL+z^d7 =sPd3 X`H^Ea+y}WIeoY=]}~H,x0aQ@z0UX&ks0. sin 33 0 obj 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 The SI unit for frequency is the hertz (Hz) and is defined as one cycle per second: 1 Hz = 1 cycle s or 1 Hz = 1 s = 1 s 1. /Name/F2 285.5 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 285.5 285.5 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 Webpoint of the double pendulum. Instead of an infinitesimally small mass at the end, there's a finite (but concentrated) lump of material. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure 16.13. endobj << /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 The most popular choice for the measure of central tendency is probably the mean (gbar). The relationship between frequency and period is. What is the most sensible value for the period of this pendulum? The quantities below that do not impact the period of the simple pendulum are.. B. length of cord and acceleration due to gravity. /BaseFont/AQLCPT+CMEX10 When is expressed in radians, the arc length in a circle is related to its radius (LL in this instance) by: For small angles, then, the expression for the restoring force is: where the force constant is given by k=mg/Lk=mg/L and the displacement is given by x=sx=s. This method for determining 805.5 896.3 870.4 935.2 870.4 935.2 0 0 870.4 736.1 703.7 703.7 1055.5 1055.5 351.8 Look at the equation below. Pendulum A is a 200-g bob that is attached to a 2-m-long string. the pendulum of the Great Clock is a physical pendulum, is not a factor that affects the period of a pendulum, Adding pennies to the pendulum of the Great Clock changes its effective length, What is the length of a seconds pendulum at a place where gravity equals the standard value of, What is the period of this same pendulum if it is moved to a location near the equator where gravity equals 9.78m/s, What is the period of this same pendulum if it is moved to a location near the north pole where gravity equals 9.83m/s. 13 0 obj Except where otherwise noted, textbooks on this site /Type/Font 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 g B. Cut a piece of a string or dental floss so that it is about 1 m long. /Subtype/Type1 351.8 935.2 578.7 578.7 935.2 896.3 850.9 870.4 915.7 818.5 786.1 941.7 896.3 442.6 << Solution: As stated in the earlier problems, the frequency of a simple pendulum is proportional to the inverse of the square root of its length namely $f \propto 1/\sqrt{\ell}$. Solve the equation I keep using for length, since that's what the question is about. 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 The pennies are not added to the pendulum bob (it's moving too fast for the pennies to stay on), but are instead placed on a small platform not far from the point of suspension. << 692.5 323.4 569.4 323.4 569.4 323.4 323.4 569.4 631 507.9 631 507.9 354.2 569.4 631 /FontDescriptor 17 0 R <> 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 . Calculate gg. /Type/Font (b) The period and frequency have an inverse relationship. 42 0 obj 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 /LastChar 196 Exams: Midterm (July 17, 2017) and . endobj endobj 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 WebQuestions & Worked Solutions For AP Physics 1 2022. 571 285.5 314 542.4 285.5 856.5 571 513.9 571 542.4 402 405.4 399.7 571 542.4 742.3 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 /BaseFont/JFGNAF+CMMI10 The time taken for one complete oscillation is called the period. There are two basic approaches to solving this problem graphically a curve fit or a linear fit. WebAnalytic solution to the pendulum equation for a given initial conditions and Exact solution for the nonlinear pendulum (also here). /LastChar 196 742.3 799.4 0 0 742.3 599.5 571 571 856.5 856.5 285.5 314 513.9 513.9 513.9 513.9 Projectile motion problems and answers Problem (1): A person kicks a ball with an initial velocity of 15\, {\rm m/s} 15m/s at an angle of 37 above the horizontal (neglect the air resistance). For the precision of the approximation 624.1 928.7 753.7 1090.7 896.3 935.2 818.5 935.2 883.3 675.9 870.4 896.3 896.3 1220.4 endobj Since the pennies are added to the top of the platform they shift the center of mass slightly upward. %PDF-1.5 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 endobj These Pendulum Charts will assist you in developing your intuitive skills and to accurately find solutions for everyday challenges. 5 0 obj If you need help, our customer service team is available 24/7. 61) Two simple pendulums A and B have equal length, but their bobs weigh 50 gf and l00 gf respectively. 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] /Type/Font 6 stars and was available to sell back to BooksRun online for the top buyback price of $ 0. 5. Here, the only forces acting on the bob are the force of gravity (i.e., the weight of the bob) and tension from the string. 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] 351.8 935.2 578.7 578.7 935.2 896.3 850.9 870.4 915.7 818.5 786.1 941.7 896.3 442.6 513.9 770.7 456.8 513.9 742.3 799.4 513.9 927.8 1042 799.4 285.5 513.9] Consider a geologist that uses a pendulum of length $35\,{\rm cm}$ and frequency of 0.841 Hz at a specific place on the Earth. Using this equation, we can find the period of a pendulum for amplitudes less than about 1515. /LastChar 196 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 277.8 500] 3.2. 896.3 896.3 740.7 351.8 611.1 351.8 611.1 351.8 351.8 611.1 675.9 546.3 675.9 546.3 Example Pendulum Problems: A. 24 0 obj 0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 If the length of the cord is increased by four times the initial length, then determine the period of the harmonic motion. /Name/F4 Students calculate the potential energy of the pendulum and predict how fast it will travel. >> Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. << << /FontDescriptor 26 0 R Current Index to Journals in Education - 1993 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 That's a gain of 3084s every 30days also close to an hour (51:24). /Name/F4 511.1 511.1 511.1 831.3 460 536.7 715.6 715.6 511.1 882.8 985 766.7 255.6 511.1] endobj 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 .p`t]>+b1Ky>%0HCW,8D/!Y6waldaZy_u1_?0-5D#0>#gb? Two simple pendulums are in two different places. endstream But the median is also appropriate for this problem (gtilde). The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. (7) describes simple harmonic motion, where x(t) is a simple sinusoidal function of time. /BaseFont/TMSMTA+CMR9 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 460 511.1 306.7 306.7 460 255.6 817.8 562.2 511.1 511.1 460 421.7 408.9 332.2 536.7 /FontDescriptor 29 0 R /Subtype/Type1 Homogeneous first-order linear partial differential equation: consent of Rice University. What is its frequency on Mars, where the acceleration of gravity is about 0.37 that on Earth? /Contents 21 0 R stream 770.7 628.1 285.5 513.9 285.5 513.9 285.5 285.5 513.9 571 456.8 571 457.2 314 513.9 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 In this case, this ball would have the greatest kinetic energy because it has the greatest speed. /Name/F1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 1 0 obj R ))jM7uM*%? 9 0 obj That means length does affect period. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663.6 885.4 826.4 736.8 Find its PE at the extreme point. As you can see, the period and frequency of a simple pendulum do not depend on the mass of the pendulum bob. If f1 is the frequency of the first pendulum and f2 is the frequency of the second pendulum, then determine the relationship between f1 and f2. /BaseFont/WLBOPZ+CMSY10 Will it gain or lose time during this movement? Energy of the Pendulum The pendulum only has gravitational potential energy, as gravity is the only force that does any work. /LastChar 196 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 Our mission is to improve educational access and learning for everyone. x a&BVX~YL&c'Zm8uh~_wsWpuhc/Nh8CQgGW[k2[6n0saYmPy>(]V@:9R+-Cpp!d::yzE q if(typeof ez_ad_units != 'undefined'){ez_ad_units.push([[300,250],'physexams_com-leader-3','ezslot_10',134,'0','0'])};__ez_fad_position('div-gpt-ad-physexams_com-leader-3-0'); Problem (11): A massive bob is held by a cord and makes a pendulum. 5 0 obj <> are not subject to the Creative Commons license and may not be reproduced without the prior and express written 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 /Type/Font (c) Frequency of a pendulum is related to its length by the following formula \begin{align*} f&=\frac{1}{2\pi}\sqrt{\frac{g}{\ell}} \\\\ 1.25&=\frac{1}{2\pi}\sqrt{\frac{9.8}{\ell}}\\\\ (2\pi\times 1.25)^2 &=\left(\sqrt{\frac{9.8}{\ell}}\right)^2 \\\\ \Rightarrow \ell&=\frac{9.8}{4\pi^2\times (1.25)^2} \\\\&=0.16\quad {\rm m}\end{align*} Thus, the length of this kind of pendulum is about 16 cm. /Name/F12 <> stream 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 Use a simple pendulum to determine the acceleration due to gravity 21 0 obj All of the methods used were appropriate to the problem and all of the calculations done were error free, so all of them.