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Ch 14: Periodic Motion
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 14: Periodic MotionProblem 15a
Chapter 14, Problem 15a

The point of the needle of a sewing machine moves in SHM along the x-axis with a frequency of 2.5 Hz. At t = 0 its position and velocity components are +1.1 cm and -15 cm/s, respectively. Find the acceleration component of the needle at t = 0.

Verified step by step guidance
1
Start by recalling the formula for simple harmonic motion (SHM) which relates acceleration, position, and angular frequency: \( a(t) = -\omega^2 x(t) \).
Determine the angular frequency \( \omega \) using the given frequency \( f = 2.5 \text{ Hz} \). The relationship is \( \omega = 2\pi f \).
Substitute the given position \( x(0) = 1.1 \text{ cm} \) into the acceleration formula \( a(t) = -\omega^2 x(t) \) to find the acceleration at \( t = 0 \).
Ensure unit consistency by converting the position from centimeters to meters if necessary, as acceleration is typically expressed in \( \text{m/s}^2 \).
Calculate the acceleration component using the values obtained from the previous steps, keeping in mind the negative sign indicates direction opposite to the displacement.

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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 the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It is characterized by sinusoidal oscillations, and key parameters include amplitude, frequency, and phase. In this problem, the needle's motion along the x-axis is described by SHM.
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Frequency and Angular Frequency

Frequency, measured in Hertz (Hz), is the number of oscillations per second. Angular frequency, denoted by ω, is related to frequency by the formula ω = 2πf, where f is the frequency. It represents the rate of change of the phase of the sinusoidal waveform, crucial for calculating other parameters like velocity and acceleration in SHM.
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Acceleration in SHM

In SHM, acceleration is given by the formula a(t) = -ω²x(t), where ω is the angular frequency and x(t) is the displacement. The negative sign indicates that acceleration is always directed towards the equilibrium position. At t = 0, knowing the position and velocity allows us to determine the acceleration using the relationship between these parameters.
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Related Practice
Textbook Question

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is 0.090 m, it takes the block 2.70 s to travel from x = 0.090 m to x = -0.090 m. If the amplitude is doubled, to 0.180 m, how long does it take the block to travel from x = 0.180 m to x = -0.180 m?

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Textbook Question

In a physics lab, you attach a 0.200-kg air-track glider to the end of an ideal spring of negligible mass and start it oscillating. The elapsed time from when the glider first moves through the equilibrium point to the second time it moves through that point is 2.60 s. Find the spring's force constant.

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Textbook Question

A 2.00-kg, frictionless block is attached to an ideal spring with force constant 300 N/m. At t = 0 the spring is neither stretched nor compressed and the block is moving in the negative direction at 12.0 m/s. Find (a) the amplitude and (b) the phase angle. (c) Write an equation for the position as a function of time.

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Textbook Question

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is 0.090 m, it takes the block 2.70 s to travel from x = 0.090 m to x = -0.090 m. If the amplitude is doubled, to 0.180 m, how long does it take the block to travel from x = 0.090 m to x = -0.090 m?

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Textbook Question

Weighing Astronauts. This procedure has been used to 'weigh' astronauts in space: A 42.5-kg chair is attached to a spring and allowed to oscillate. When it is empty, the chair takes 1.30 s to make one complete vibration. But with an astronaut sitting in it, with her feet off the floor, the chair takes 2.54 s for one cycle. What is the mass of the astronaut?

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Textbook Question

An object is undergoing SHM with period 0.900 s and amplitude 0.320 m. At t = 0 the object is at x = 0.320 m and is instantaneously at rest. Calculate the time it takes the object to go (a) from x = 0.320 m to x = 0.160 m. (b) from x = 0.160 m to x = 0.

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