Textbook Question
A certain simple pendulum has a period on the earth of 1.60 s. What is its period on the surface of Mars, where g = 3.71 m/s2?
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Ch 14: Periodic Motion
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors37
- Ch 02: Motion Along a Straight Line57
- Ch 03: Motion in Two or Three Dimensions45
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws54
- Ch 06: Work & Kinetic Energy11
- Ch 07: Potential Energy & Conservation6
- Ch 08: Momentum, Impulse, and Collisions15
- Ch 09: Rotation of Rigid Bodies37
- Ch 10: Dynamics of Rotational Motion44
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics27
- Ch 13: Gravitation34
- Ch 14: Periodic Motion26
- Ch 15: Mechanical Waves22
- Ch 16: Sound & Hearing28
- Ch 17: Temperature and Heat28
- Ch 18: Thermal Properties of Matter35
- Ch 19: The First Law of Thermodynamics29
- Ch 20: The Second Law of Thermodynamics18
- Ch 21: Electric Charge and Electric Field28
- Ch 22: Gauss' Law21
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF24
- Ch 26: Direct-Current Circuits29
- Ch 27: Magnetic Field and Magnetic Forces16
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction22
- Ch 30: Inductance14
- Ch 31: Alternating Current20
- Ch 33: The Nature and Propagation of Light17
- Ch 34: Geometric Optics29
- Ch 35: Interference13
- Ch 36: Diffraction19
- Ch 37: Special Relativity19
- Ch 38: Photons: Light Waves Behaving as Particles12
- Ch 39: Particles Behaving as Waves25
- Ch 40: Quantum Mechanics I: Wave Functions20
- Ch 41: Quantum Mechanics II: Atomic Structure21
- Ch 42: Molecules and Condensed Matter17
- Ch 43: Nuclear Physics13
- Ch 44: Particle Physics and Cosmology17
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
Chapter 14, Problem 43b
You pull a simple pendulum 0.240 m long to the side through an angle of 3.50° and release it. How much time does it take if the pendulum is released at an angle of 1.75° instead of 3.50°?
Verified step by step guidance1
Understand that the period of a simple pendulum is given by the formula: , where is the length of the pendulum and is the acceleration due to gravity.
Note that the period is independent of the amplitude (angle) for small angles, which is a characteristic of simple harmonic motion. Therefore, the angle change from 3.50° to 1.75° does not affect the period.
Calculate the period using the given length of the pendulum, , and the standard value of gravitational acceleration, .
Substitute the values into the formula: .
Simplify the expression to find the period , which is the time it takes for one complete oscillation of the pendulum.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Simple Pendulum
A simple pendulum consists of a mass (called the bob) attached to a string or rod of fixed length, which swings back and forth under the influence of gravity. The motion is periodic and can be approximated as simple harmonic motion for small angles, where the restoring force is proportional to the displacement angle.
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Simple Harmonic Motion of Pendulums
Period of a Pendulum
The period of a pendulum is the time it takes to complete one full cycle of its motion. For small angles, the period is approximately independent of the amplitude and is given by the formula T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. This approximation holds true for angles less than about 15°.
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Small Angle Approximation
The small angle approximation is used in pendulum motion to simplify calculations by assuming that sin(θ) ≈ θ when θ is measured in radians and is small. This allows the pendulum's motion to be treated as simple harmonic, making the period formula applicable. This approximation is valid for angles typically less than 15°, ensuring the pendulum behaves predictably.
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Related Practice
Textbook Question
You pull a simple pendulum 0.240 m long to the side through an angle of 3.50° and release it. How much time does it take the pendulum bob to reach its highest speed?
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Textbook Question
A building in San Francisco has light fixtures consisting of small 2.35-kg bulbs with shades hanging from the ceiling at the end of light, thin cords 1.50 m long. If a minor earthquake occurs, how many swings per second will these fixtures make?
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Textbook Question
A thrill-seeking cat with mass 4.00 kg is attached by a harness to an ideal spring of negligible mass and oscillates vertically in SHM. The amplitude is 0.050 m, and at the highest point of the motion the spring has its natural unstretched length. Calculate the elastic potential energy of the spring (take it to be zero for the unstretched spring), the kinetic energy of the cat, the gravitational potential energy of the system relative to the lowest point of the motion, and the sum of these three energies when the cat is at its highest point.
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Textbook Question
A mass is oscillating with amplitude A at the end of a spring. How far (in terms of A) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic energy?
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