- Accounting
- Aerospace Engineering
- Anatomy
- Anthropology
- Arts & Humanities
- Astronomy
- Biology
- Business
- Chemistry
- Civil Engineering
- Computer Science
- Communications
- Economics
- Electrical Engineering
- English
- Finance
- Geography
- Geology
- Health Science
- History
- Industrial Engineering
- Information Systems
- Law
- Linguistics
- Management
- Marketing
- Material Science
- Mathematics
- Mechanical Engineering
- Medicine
- Nursing
- Philosophy
- Physics
- Political Science
- Psychology
- Religion
- Sociology
- Statistics

HomeStudy GuidesPhysics

Menu

The Nature of Science and Physics

Kinematics

Introduction to One-Dimensional KinematicsDisplacementVectors, Scalars, and Coordinate SystemsTime, Velocity, and SpeedVideo: One-Dimensional KinematicsAccelerationMotion Equations for Constant Acceleration in One DimensionProblem-Solving Basics for One-Dimensional KinematicsFalling ObjectsGraphical Analysis of One-Dimensional Motion

Two-Dimensional Kinematics

Dynamics: Force and Newton's Laws of Motion

Introduction to Dynamics: Newton's Laws of MotionDevelopment of Force ConceptNewton's First Law of Motion: InertiaNewton's Second Law of Motion: Concept of a SystemNewton's Third Law of Motion: Symmetry in ForcesVideo: Newton's LawsNormal, Tension, and Other Examples of ForcesProblem-Solving StrategiesFurther Applications of Newton's Laws of MotionExtended Topic: The Four Basic Forces—An Introduction

Further Applications of Newton's Laws: Friction, Drag, and Elasticity

Uniform Circular Motion and Gravitation

Introduction to Uniform Circular Motion and GravitationRotation Angle and Angular VelocityCentripetal AccelerationCentripetal ForceFictitious Forces and Non-inertial Frames: The Coriolis ForceNewton's Universal Law of GravitationVideo: GravitationSatellites and Kepler's Laws: An Argument for Simplicity

Work, Energy, and Energy Resources

Introduction to Work, Energy, and Energy ResourcesWork: The Scientific DefinitionKinetic Energy and the Work-Energy TheoremGravitational Potential EnergyVideo: Potential and Kinetic EnergyConservative Forces and Potential EnergyNonconservative ForcesConservation of EnergyPowerWork, Energy, and Power in HumansWorld Energy Use

Linear Momentum and Collisions

Rotational Motion and Angular Momentum

Introduction to Rotational Motion and Angular MomentumAngular AccelerationKinematics of Rotational MotionVideo: Rotational MotionDynamics of Rotational Motion: Rotational InertiaRotational Kinetic Energy: Work and Energy RevisitedAngular Momentum and Its ConservationVideo: Angular MomentumCollisions of Extended Bodies in Two DimensionsGyroscopic Effects: Vector Aspects of Angular Momentum

Statics and Torque

Fluid Statics

Introduction to Fluid StaticsWhat Is a Fluid?DensityPressureVariation of Pressure with Depth in a FluidPascal's PrincipleGauge Pressure, Absolute Pressure, and Pressure MeasurementArchimedes' PrincipleVideo: BuoyancyCohesion and Adhesion in Liquids: Surface Tension and Capillary ActionPressures in the Body

Fluid Dynamics and Its Biological and Medical Applications

Introduction to Fluid Dynamics and Biological and Medical ApplicationsFlow Rate and Its Relation to VelocityBernoulli's EquationVideo: Fluid FlowThe Most General Applications of Bernoulli's EquationViscosity and Laminar Flow; Poiseuille's LawThe Onset of TurbulenceMotion of an Object in a Viscous FluidMolecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

Temperature, Kinetic Theory, and the Gas Laws

Heat and Heat Transfer Methods

Thermodynamics

Introduction to ThermodynamicsThe First Law of ThermodynamicsThe First Law of Thermodynamics and Some Simple ProcessesIntroduction to the Second Law of Thermodynamics: Heat Engines and Their EfficiencyCarnot's Perfect Heat Engine: The Second Law of Thermodynamics RestatedApplications of Thermodynamics: Heat Pumps and RefrigeratorsEntropy and the Second Law of Thermodynamics: Disorder and the Unavailability of EnergyStatistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

Oscillatory Motion and Waves

Introduction to Oscillatory Motion and WavesHooke's Law: Stress and Strain RevisitedPeriod and Frequency in OscillationsSimple Harmonic Motion: A Special Periodic MotionVideo: Harmonic MotionThe Simple PendulumEnergy and the Simple Harmonic OscillatorUniform Circular Motion and Simple Harmonic MotionDamped Harmonic MotionForced Oscillations and ResonanceWavesSuperposition and InterferenceEnergy in Waves: Intensity

Physics of Hearing

- Measure acceleration due to gravity.

In Figure 1 we see that 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. The linear displacement from equilibrium is

Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A

We begin by defining the displacement to be the arc length

Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about 15º), sin

*F* ≈ −*mg**θ*.

$\theta=\frac{s}{L}\\$

.
For small angles, then, the expression for the restoring force is:$F\approx-\frac{mg}{L}s\\$

.
This expression is of the form: $k=\frac{mg}{L}\\$

and the displacement is given by Using this equation, we can find the period of a pendulum for amplitudes less than about 15º. For the simple pendulum:

$\displaystyle{T}=2\pi\sqrt{\frac{m}{k}}=2\pi\sqrt{\frac{m}{\frac{mg}{L}}}\\$

Thus, $T=2\pi\sqrt{\frac{L}{g}}\\$

for the period of a simple pendulum. This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. As with simple harmonic oscillators, the period Note the dependence of

$T=2\pi\sqrt{\frac{L}{g}}\\$

for $T=2\pi\sqrt{\frac{L}{g}}\\$

and solve for $g=4\pi^{2}\frac{L}{T^{2}}\\$

.
Substitute known values into the new equation:$g=4\pi^{2}\frac{0.750000\text{ m}}{\left(1.7357\text{ s}\right)^{2}}\\$

.
Calculate to find *g* = 9.8281 m/s^{2}.

- A mass
*m*suspended by a wire of length*L*is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about 15º. - The period of a simple pendulum is $T=2\pi\sqrt{\frac{L}{g}}\\$, where
*L*is the length of the string and*g*is the acceleration due to gravity.

- Pendulum clocks are made to run at the correct rate by adjusting the pendulum’s length. Suppose you move from one city to another where the acceleration due to gravity is slightly greater, taking your pendulum clock with you, will you have to lengthen or shorten the pendulum to keep the correct time, other factors remaining constant? Explain your answer.

- What is the length of a pendulum that has a period of 0.500 s?
- Some people think a pendulum with a period of 1.00 s can be driven with "mental energy" or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?
- What is the period of a 1.00-m-long pendulum?
- How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot?
- The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency?
- Two parakeets sit on a swing with their combined center of mass 10.0 cm below the pivot. At what frequency do they swing?
- (a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is 9.79 m/s
^{2}is moved to a location where it the acceleration due to gravity is 9.82 m/s^{2}. What is its new period? (b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity. - A pendulum with a period of 2.00000 s in one location (
*g*= 9.80 m/s^{2}) is moved to a new location where the period is now 1.99796 s. What is the acceleration due to gravity at its new location? - (a) What is the effect on the period of a pendulum if you double its length? (b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?
- Find the ratio of the new/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is 1.63 m/s
^{2}. - At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is 1.63 m/s
^{2}, if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock’s hour hand to make one revolution on the Moon. - Suppose the length of a clock’s pendulum is changed by 1.000%, exactly at noon one day. What time will it read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.
- If a pendulum-driven clock gains 5.00 s/day, what fractional change in pendulum length must be made for it to keep perfect time?

3. 2.01 s

5. 2.23 Hz

7. (a) 2.99541 s; (b) Since the period is related to the square root of the acceleration of gravity, when the acceleration changes by 1% the period changes by (0.01)

9. (a) Period increases by a factor of 1.41

$\left(\sqrt{2}\right)\\$

; (b) Period decreases to 97.5% of old period11. Slow by a factor of 2.45

13. length must increase by 0.0116%