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Chapter 15: Q. 38 (page 416)

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

Expert verified

Part a: The fraction of potential energy is 14Eand the fraction of kinetic energy is 34E.

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

Step by step solution

01

a. Introduction

The Potential Energy of Mass on spring U=12kx2--- (1)

The Kinetic Energy of Mass on spring K=12kA2 ---(2)

From the question, we know that

Displacementx=12A---(3).

02

b. Calculation

Let us substitute (3) in (1).

U=12Kx2

U=12K12A2U=1412KA2

U=14E

Therefore

Kinetic Energy K = Total Mechanical Energy 鈥� Potential Energy

K=E鈭�14E

K=34E.

03

b. Introduction

Assume y is the fraction of A

Displacement X = yA.

04

b. Calculation

As we know,

Energy = Potential Energy + Kinetic Energy

ETotal=U+K

i.e.,

ETotal=12kx2+12kA212kA2=1212kx2

A2=12(yA)2(as we knowx=yA)

A2=12y

y=2

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

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