Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Chapter 16, Problem 4PQ

(a)

To determine

The position, velocity, and acceleration of simple harmonic oscillator at time t=0.

(a)

Expert Solution
Check Mark

Answer to Problem 4PQ

The position, velocity, and acceleration of simple harmonic oscillator at time t=0 is 0.398m, 7.81m/s, and 43.1m/s2 respectively.

Explanation of Solution

Write the equation to find the position of simple harmonic oscillator.

    y(t)=(0.850m)cos(10.4t5.20)                                                                           (I)

Here, y(t) is the position at time t

Differentiate the above expression with respect to t to find the velocity.

  v(t)=d((0.850m)cos(10.4t5.20))dt=(0.850m)(10.4rad/s)sin(10.4t5.20)                                                         (II)

Here, v(t) is the velocity of particle at time t.

Differentiate the above expression with respect to t to find the acceleration.

  a(t)=d((0.850m)(10.4rad/s)sin(10.4t5.20))dt=(10.4rad/s)2(0.850m)cos(10.4t5.20)                                                 (III)

Conclusion:

Substitute 0s for t in equation (I) to find y(0).

    y(0)=(0.850m)cos(10.4(0s)5.20)=(0.850m)(0.468)=0.398m

Substitute 0s for t in equation (II) to find v(0).

    v(0)=(0.850m)(10.4rad/s)sin(10.4(0s)5.20)=(8.84m/s)(0.883)=7.81m/s

Substitute 0s for t in equation (III) to find a(0).

    a(0)=(10.4rad/s)2(0.850m)cos(10.4(0s)5.20)=91.9m/s2(0.469)=43.1m/s2

Therefore, The position, velocity, and acceleration of simple harmonic oscillator at time t=0 is 0.398m, 7.81m/s, and 43.1m/s2 respectively.

(b)

To determine

The position, velocity, and acceleration of simple harmonic oscillator at time t=0.500s.

(b)

Expert Solution
Check Mark

Answer to Problem 4PQ

The position, velocity, and acceleration of simple harmonic oscillator at time t=0.500s is 0.850m, 0m/s, and 91.9m/s2 respectively.

Explanation of Solution

Substitute 0.500s for t in equation (I) to find y(0).

    y(0)=(0.850m)cos(10.4(0.500s)5.20)=(0.850m)(1)=0.850m

Substitute 0.500s for t in equation (II) to find v(0).

    v(0)=(0.850m)(10.4rad/s)sin(10.4(0.500s)5.20)=(8.84m/s)(0)=0m/s

Substitute 0.500s for t in equation (III) to find a(0).

    a(0)=(10.4rad/s)2(0.850m)cos(10.4(0.500s)5.20)=91.9m/s2(1)=91.9m/s2

Conclusion:

Therefore, the position, velocity, and acceleration of simple harmonic oscillator at time t=0.500s is 0.850m, 0m/s, and 91.9m/s2 respectively.

(c)

To determine

The position, velocity, and acceleration of simple harmonic oscillator at time t=2.00s.

(c)

Expert Solution
Check Mark

Answer to Problem 4PQ

The position, velocity, and acceleration of simple harmonic oscillator at time t=2.00s is 0.845m, 0.953m/s, and 91.4m/s2 respectively.

Explanation of Solution

Substitute 2.00s for t in equation (I) to find y(0).

    y(0)=(0.850m)cos(10.4(2.00s)5.20)=(0.850m)(0.994)=0.845m

Substitute 2.00s for t in equation (II) to find v(0).

    v(0)=(0.850m)(10.4rad/s)sin(10.4(2.00s)5.20)=(8.84m/s)(0.107)=0.953m/s

Substitute 2.00s for t in equation (III) to find a(0).

    a(0)=(10.4rad/s)2(0.850m)cos(10.4(2.00s)5.20)=91.9m/s2(0.994)=91.4m/s2

Conclusion:

Therefore, the position, velocity, and acceleration of simple harmonic oscillator at time t=2.00s is 0.845m, 0.953m/s, and 91.4m/s2 respectively.

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Chapter 16 Solutions

Physics for Scientists and Engineers: Foundations and Connections

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