Question

At what displacement, as a fraction of a, is the mechanical energy half kinetic and half potential?

Answer 1

The displacement is ¹/₂√2 A

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Further explanation

Simple Harmonic Motion is a motion where the magnitude of acceleration is directly proportional to the magnitude of the displacement but in the opposite direction.

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The pulled and then released spring is one of the examples of Simple Harmonic Motion. We can use the following formula to find the period of this spring.

$\large {\boxed {T = 2 \pi\sqrt{\frac{m}{k}}}}$

T = Periode of Spring ( second )

m = Load Mass ( kg )

k = Spring Constant ( N / m )

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The pendulum which moves back and forth is also an example of Simple Harmonic Motion. We can use the following formula to find the period of this pendulum.

$\large {\boxed {T = 2 \pi\sqrt{\frac{L}{g}}}}$

T = Periode of Pendulum ( second )

L = Length of Pendulum ( kg )

g = Gravitational Acceleration ( m/s² )

Let us now tackle the problem !

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Given:

amplitude of motion = A

kinetic energy(K) = potential energy(U)

Asked:

displacement = x = ?

Solution:

We will use conservation of energy as follows:

$U_{max} = U + K$

$U_{max} = U + U$

$U_{max} = 2U$

$\frac{1}{2}k A^2 = 2 \times \frac{1}{2}kx^2$

$A^2 = 2x^2$

$x^2 = \frac{1}{2}A^2$

$x = \sqrt{\frac{1}{2}A^2}$

$\large {\boxed {x = \frac{1}{2} \sqrt{2} A}}$

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Learn more

• Model for Simple Harmonic Motion : https://brainly.com/question/9221526
• Force of Simple Harmonic Motion : https://brainly.com/question/3323600
• Example of Simple Harmonic Motion : https://brainly.com/question/11892568

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Answer details

Grade: High School

Subject: Physics

Chapter: Simple Harmonic Motion

Answer 2

The displacement  will be given by  ¹/₂√2 A

What is simple harmonic motion?

Simple Harmonic Motion is a motion where the magnitude of acceleration is directly proportional to the magnitude of the displacement but in the opposite direction.

The pulled and then released spring is one of the examples of Simple Harmonic Motion. We can use the following formula to find the period of this spring.

$T=2\pi\sqrt{\dfrac{m}{k}}$

T = Periode of Spring ( second )

m = Load Mass ( kg )

k = Spring Constant ( N / m )

The pendulum which moves back and forth is also an example of Simple Harmonic Motion. We can use the following formula to find the period of this pendulum.

$T=2\pi\sqrt{\dfrac{L}{g}}$

T = Periode of Pendulum ( second )

L = Length of Pendulum ( kg )

g = Gravitational Acceleration ( m/s² )

Let us now tackle the problem !

Given:

amplitude of motion = A

kinetic energy(K) = potential energy(U)

displacement = x = ?

We will use conservation of energy as follows:

$U_{max}=U+K$

$U_{max}=U+U$

$U_{max}=2U$

$\dfrac{1}{2}kA^2=2\times \dfrac{1}{2}kx^2$

$A^2=2x^2$

$x^2=\dfrac{1}{2}A^2$

$x=\sqrt{\dfrac{1}{2}A^2$

$x=\dfrac{1}{2}\sqrt2A$

Hence the displacement  will be given by  ¹/₂√2 A

To know more about Simple harmonic motion follow

https://brainly.com/question/17315536