Domanda del libro di testo
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 Calc14th EditionUniversity PhysicsISBN: 9780321973610
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TOC.tableOfContents
- Ch 01: Units, Physical Quantities & Vectors41
- Ch 02: Motion Along a Straight Line67
- Ch 03: Motion in Two or Three Dimensions47
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws59
- Ch 06: Work & Kinetic Energy14
- Ch 07: Potential Energy & Conservation8
- Ch 08: Momentum, Impulse, and Collisions18
- Ch 09: Rotation of Rigid Bodies42
- Ch 10: Dynamics of Rotational Motion48
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics31
- Ch 13: Gravitation35
- Ch 14: Periodic Motion32
- Ch 15: Mechanical Waves25
- Ch 16: Sound & Hearing32
- Ch 17: Temperature and Heat36
- Ch 18: Thermal Properties of Matter38
- Ch 19: The First Law of Thermodynamics33
- Ch 20: The Second Law of Thermodynamics22
- Ch 21: Electric Charge and Electric Field36
- Ch 22: Gauss' Law24
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF26
- Ch 26: Direct-Current Circuits30
- Ch 27: Magnetic Field and Magnetic Forces18
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction28
- Ch 30: Inductance17
- Ch 31: Alternating Current21
- Ch 32: Electromagnetic Waves1
- Ch 33: The Nature and Propagation of Light20
- Ch 34: Geometric Optics34
- Ch 35: Interference19
- Ch 36: Diffraction25
- Ch 37: Special Relativity20
- Ch 38: Photons: Light Waves Behaving as Particles15
- Ch 39: Particles Behaving as Waves26
- Ch 40: Quantum Mechanics I: Wave Functions21
- Ch 41: Quantum Mechanics II: Atomic Structure25
- Ch 42: Molecules and Condensed Matter18
- Ch 43: Nuclear Physics16
- Ch 44: Particle Physics and Cosmology23
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Capitolo 14, Problema 51ab
A simple pendulum 2.00 m long swings through a maximum angle of 30.0° with the vertical. Calculate its period (a) assuming a small amplitude, and (b) using the first three terms of Eq. (14.35).
Guida verificata passo dopo passo1
Step 1: For part (a), use the formula for the period of a simple pendulum with small amplitude: , where is the length of the pendulum and is the acceleration due to gravity (approximately 9.81 m/s²). Substitute m into the formula.
Step 2: For part (b), use the first three terms of the series expansion for the period of a pendulum with larger amplitude: , where is the maximum angle in radians. Convert 30.0° to radians using .
Step 3: Calculate the period using the formula from step 1 for small amplitude and the formula from step 2 for larger amplitude. Compare the two results to determine which is more accurate.
Step 4: To find the percentage error, use the formula: , where is the period calculated assuming small amplitude and is the period calculated using the series expansion.
Step 5: Interpret the results: The more accurate answer is the one that accounts for the larger amplitude using the series expansion. Calculate the percentage error to quantify the difference between the two methods.
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Simple Harmonic Motion
Simple harmonic motion (SHM) describes the motion of oscillating systems like pendulums, where the restoring force is directly proportional to the displacement. For small angles, a simple pendulum exhibits SHM, allowing us to use the formula T = 2π√(L/g) to calculate its period, where L is the length and g is the acceleration due to gravity.
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07:52
Simple Harmonic Motion of Pendulums
Pendulum Period Formula
The period of a simple pendulum for small angles is given by T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. This formula assumes the angle of swing is small enough for the approximation sin(θ) ≈ θ to hold, simplifying the motion to SHM.
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06:53Energy in Pendulums
Nonlinear Pendulum Motion
For larger angles, the simple pendulum does not follow simple harmonic motion exactly, and corrections are needed. The period can be calculated using a series expansion, such as Eq. (14.35), which includes higher-order terms to account for the nonlinearity. This provides a more accurate period by considering deviations from the small-angle approximation.
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07:52Simple Harmonic Motion of Pendulums
Pratica correlata
Domanda del libro di testo
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°?
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Domanda del libro di testo
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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Domanda del libro di testo
A simple pendulum 2.00 m long swings through a maximum angle of 30.0° with the vertical. Calculate its period (a) assuming a small amplitude, and (b) using the first three terms of Eq. (14.35). (c) Which of the answers in parts (a) and (b) is more accurate? What is the percentage error of the less accurate answer compared with the more accurate one?
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