Textbook Question
A uniform rod of length L oscillates as a pendulum about a pivot that is a distance x from the center.
a. For what value of x, in terms of L, is the oscillation period a minimum?
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Ch 15: Oscillations
Chapter 15, Problem 15
FIGURE EX15.7 is the position-versus-time graph of a particle in simple harmonic motion.
c. What is vₘₐₓ?
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Identify the amplitude (A) of the simple harmonic motion from the graph. The amplitude is the maximum displacement from the equilibrium position, which is 5 cm.
Determine the period (T) of the motion. The period is the time it takes for one complete cycle of motion. From the graph, one complete cycle occurs from t = 0 s to t = 4 s, so T = 4 s.
Calculate the angular frequency (ω) using the formula ω = 2π/T. Substitute the value of T to find ω.
Use the formula for maximum velocity in simple harmonic motion, v_max = Aω. Substitute the values of A and ω to find v_max.
Express the final formula for v_max in terms of the given values and constants.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Simple Harmonic Motion (SHM)
Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion is characterized by a restoring force proportional to the displacement from the equilibrium, leading to a sinusoidal position versus time graph. In SHM, the maximum displacement from the equilibrium is called amplitude, and the motion is defined by parameters such as frequency and period.
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Maximum Velocity (vₘₐₓ)
The maximum velocity (vₘₐₓ) in simple harmonic motion occurs as the object passes through the equilibrium position. It can be calculated using the formula vₘₐₓ = ωA, where ω is the angular frequency and A is the amplitude of the motion. This velocity is crucial for understanding the dynamics of the oscillating system and indicates the highest speed the particle achieves during its motion.
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Position vs. Time Graph
A position versus time graph visually represents the motion of an object over time. In the context of simple harmonic motion, this graph typically shows a sinusoidal pattern, indicating periodic oscillations. The peaks of the graph correspond to the maximum displacement (amplitude), while the zero crossings indicate the moments when the object passes through the equilibrium position, where its velocity is at its maximum.
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Related Practice
Textbook Question
An object in simple harmonic motion has an amplitude of 8.0 cm, n angular frequency of 0.25 rad/s, and a phase constant of π rad. Draw a velocity graph showing two cycles of the motion.
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Textbook Question
The amplitude of an oscillator decreases to 36.8% of its initial value in 10.0 s. What is the value of the time constant?
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Textbook Question
An object in SHM oscillates with a period of 4.0 s and an amplitude of 10 cm. How long does the object take to move from x = 0.0 cm to x = 6.0 cm?
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Textbook Question
A 500 g wood block on a frictionless table is attached to a horizontal spring. A 50 g dart is shot into the face of the block opposite the spring, where it sticks. Afterward, the spring oscillates with a period of 1.5 s and an amplitude of 20 cm. How fast was the dart moving when it hit the block?
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Textbook Question
A 350 g mass on a 45-cm-long string is released at an angle of 4.5° from vertical. It has a damping constant of 0.010 kg/s. After 25 s, (a) how many oscillations has it completed and (b) what fraction of the initial energy has been lost?
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