Chapter 15: Q. 38 (page 416)

a. When the displacement of a mass on a spring is $\frac{1}{2} A$ , what fraction of the energy is kinetic energy and what fraction is potential energy?
b. At what displacement, as a fraction of A, is the energy half kinetic and half potential?

Short Answer

Expert verified

Part a: The fraction of potential energy is $\frac{1}{4} E$ and the fraction of kinetic energy is $\frac{3}{4} E$ .

Part b: The energy half kinetic and half potential at displacement is $x = 2 A$ .

Step by step solution

a. Introduction

The Potential Energy of Mass on spring $U = \frac{1}{2} k x^{2}$ --- (1)

The Kinetic Energy of Mass on spring $K = \frac{1}{2} k A^{2}$ ---(2)

From the question, we know that

Displacement $x = \frac{1}{2} A$ ---(3).

b. Calculation

Let us substitute (3) in (1).

$U = \frac{1}{2} K x^{2}$

$\begin{aligned} U = \frac{1}{2} K {(\frac{1}{2} A)}^{2} \\ U = \frac{1}{4} (\frac{1}{2} K A^{2}) \end{aligned}$

$U = \frac{1}{4} E$

Therefore

Kinetic Energy K = Total Mechanical Energy – Potential Energy

$K = E - \frac{1}{4} E$

$K = \frac{3}{4} E$ .

b. Introduction

Assume y is the fraction of A

Displacement X = yA.

b. Calculation

As we know,

Energy = Potential Energy + Kinetic Energy

$E_{Total} = U + K$

i.e.,

$\begin{matrix} E_{Total} = \frac{1}{2} k x^{2} + \frac{1}{2} k A^{2} \\ \frac{1}{2} k A^{2} = \frac{1}{2} (\frac{1}{2} k x^{2}) \end{matrix}$

$A^{2} = \frac{1}{2} (y A)^{2} (as we know x = y A)$

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

$y = 2$

Therefore at displacement $x = 2 A$ , the energy will be half Kinetic and half Potential.

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a. When the displacement of a mass on a spring is $\frac{1}{2} A$ , what fraction of the energy is kinetic energy and what fraction is potential energy?
b. At what displacement, as a fraction of A, is the energy half kinetic and half potential?

Short Answer

Step by step solution

a. Introduction

b. Calculation

b. Introduction

b. Calculation

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a. When the displacement of a mass on a spring is 12A, what fraction of the energy is kinetic energy and what fraction is potential energy?b. At what displacement, as a fraction of A, is the energy half kinetic and half potential?

Short Answer

Step by step solution

a. Introduction

b. Calculation

b. Introduction

b. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Modern Physics

Electric Charge Field and Potential

Conservation of Energy and Momentum

Rotational Dynamics

Quantum Physics

Radiation

Study anywhere. Anytime. Across all devices.

a. When the displacement of a mass on a spring is $\frac{1}{2} A$ , what fraction of the energy is kinetic energy and what fraction is potential energy?
b. At what displacement, as a fraction of A, is the energy half kinetic and half potential?