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Q. 16

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

A mass attached to a horizontal spring oscillates at a frequency of . At , the mass is at and has . Determine:

The period.

The angular frequency.

The amplitude.

The phase constant.

The maximum speed.

The maximum acceleration.

The total energy.

The position at .

The period is

The angular frequency is

The amplitude is

The phase constant is

The maximum speed is

The maximum acceleration is

The total energy is

The position at is

See the step by step solution

Step by Step Solution

Step 1 Given.

Given

Required .

Step 2 Determining Period.

The period is given by

Step 3 Finding angular frequency.

Angular frequency as a function in frequency is

Substitution in yields

Step 4 Finding Amplitude.

To get the amplitude we need to find the spring constant which is given by

So use conservation's law

Use conditions to get that

Step 5 To find phase constant.

To get phase constant we want to use initial condition but let's start with equation

Apply Condition to get that

The displacement is positive at is positive and the angle may be in or quadratic where cos is positive, to determine which value s correct we need to determine it by comparing it with velocity .

Velocity is a sin wave which is positive in and quadratic, negative in and quadratic and we know that phase constant is in or quadratic so if the argument is in quadratic the argument of sin will be positive and the velocity will be negative because of - sign due to differentiation but if the argument is in Lth quadratic it will be negative and the velocity will be positive so the right answer is

Step 6 Finding maximum speed.

Speed is the differentiation of displacement with respect to time

The max positive speed is

Step 7 To find maximum acceleration.

Acceleration is differentiation of speed with respect to times

SO the maximum acceleration is given by

Step 8 To find Total energy.

The energy stored in the system is given by

Step 9 Finding the position When .

Substitution in yields

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