Ch 14: Periodic Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 14: Periodic Motion

Problem 11

Chapter 14, Problem 11

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.

Verified step by step guidance

Step 1: Understand the problem. The object is undergoing Simple Harmonic Motion (SHM) with a given period and amplitude. At t = 0, the object is at its maximum displacement (amplitude) and is momentarily at rest. We need to calculate the time taken for the object to move between specified positions.

Step 2: Recall the equation for SHM. The displacement x as a function of time t can be expressed as:

x = A \cos (ω t + φ)

, where A is the amplitude,

ω

is the angular frequency, and

φ

is the phase constant.

Step 3: Calculate the angular frequency

ω

. The angular frequency is related to the period T by the formula:

ω = \frac{2 π}{T}

. Substitute the given period to find

ω

Step 4: Determine the phase constant

φ

. Since the object is at rest at its maximum displacement at t = 0, the phase constant

φ

is 0, because

\cos (φ)

must equal 1.

Step 5: Solve for the time intervals. (a) Use the SHM equation to find the time when the object is at x = 0.160 m. Set

x

to 0.160 m and solve for t. (b) Repeat the process for x = 0, starting from x = 0.160 m. Use the SHM equation to find the time taken for this interval.

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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, with parameters such as amplitude, period, and frequency defining the motion's specifics.

Amplitude

Amplitude in SHM refers to the maximum displacement of the object from its equilibrium position. It is a measure of the energy in the system and determines the extent of oscillation. In this problem, the amplitude is 0.320 m, indicating the furthest point the object reaches from the center of its motion.

Period

The period of SHM is the time taken for one complete cycle of motion. It is inversely related to frequency and is a crucial factor in determining the timing of oscillations. Here, the period is 0.900 s, which helps calculate the time intervals for specific displacements during the object's motion.

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Related Practice

Textbook Question

A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7 shows a graph of the ball's position x as a function of time t. What are the oscillation's (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

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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 = 0.180 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.

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

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.

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