
The amplitude of a damped oscillator decreases to 0.9 times its original magnitude is 5 seconds. In another 10 seconds it will decrease to ‘a’ times its original magnitude where $\alpha $ equals
A. 0.7
B. 0.81
C. 0.729
D. 0.6
Answer
164.1k+ views
Hint: In case of damped oscillations, first try to find the relation between the amplitude at time t and the initial amplitude. Then put the values of both the amplitude and the time t is given and finally using that time value find the value of amplitude after 10 sec time.
Formula used
$A = {A_0}{e^{ - \alpha t}}$
Where, A is the amplitude at a given time t.
And ${A_0}$is the initial amplitude.
Complete answer:
For case 1: t = 5 seconds
Given the amplitude at time t = 5 seconds becomes 0.9 times of its initial amplitude.
$A = 0.9{A_0}$
Putting this value in the formula, we get;
$0.9{A_0} = {A_0}{e^{ - 5\alpha }}$
After solving, we get: ${e^{ - 5\alpha }} = 0.9$ (equation 1)
For case 2: t= 10 seconds
$a{A_0} = {A_0}{e^{ - 10\alpha }}$
After solving the above equation, we get;
${e^{ - 10\alpha }} = a$ (equation 2)
Solving equation 1 and 2, we get;
${e^{ - 10\alpha }} = a = {({e^{ - 5\alpha }})^2}$
$a = {(0.9)^2} = 0.81$
${e^{ - 10\alpha }} = 0.81$
After solving, we get;
$\alpha = 0.729$
Hence, the correct answer is Option C.
Note:Be careful about the change in amplitude according to the time given and how many times it becomes of the initial amplitude or the maximum amplitude. Use the same formula for both the case at t = 5 seconds and t = 10 seconds. Finally solving both cases together gets the value of a.
Formula used
$A = {A_0}{e^{ - \alpha t}}$
Where, A is the amplitude at a given time t.
And ${A_0}$is the initial amplitude.
Complete answer:
For case 1: t = 5 seconds
Given the amplitude at time t = 5 seconds becomes 0.9 times of its initial amplitude.
$A = 0.9{A_0}$
Putting this value in the formula, we get;
$0.9{A_0} = {A_0}{e^{ - 5\alpha }}$
After solving, we get: ${e^{ - 5\alpha }} = 0.9$ (equation 1)
For case 2: t= 10 seconds
$a{A_0} = {A_0}{e^{ - 10\alpha }}$
After solving the above equation, we get;
${e^{ - 10\alpha }} = a$ (equation 2)
Solving equation 1 and 2, we get;
${e^{ - 10\alpha }} = a = {({e^{ - 5\alpha }})^2}$
$a = {(0.9)^2} = 0.81$
${e^{ - 10\alpha }} = 0.81$
After solving, we get;
$\alpha = 0.729$
Hence, the correct answer is Option C.
Note:Be careful about the change in amplitude according to the time given and how many times it becomes of the initial amplitude or the maximum amplitude. Use the same formula for both the case at t = 5 seconds and t = 10 seconds. Finally solving both cases together gets the value of a.
Recently Updated Pages
JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Main 2025 Session 2: Exam Date, Admit Card, Syllabus, & More

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
Degree of Dissociation and Its Formula With Solved Example for JEE

Charging and Discharging of Capacitor

Instantaneous Velocity - Formula based Examples for JEE

JEE Main Chemistry Question Paper with Answer Keys and Solutions

JEE Main Reservation Criteria 2025: SC, ST, EWS, and PwD Candidates

What is Normality in Chemistry?

Other Pages
Total MBBS Seats in India 2025: Government College Seat Matrix

NEET Total Marks 2025: Important Information and Key Updates

Neet Cut Off 2025 for MBBS in Tamilnadu: AIQ & State Quota Analysis

Karnataka NEET Cut off 2025 - Category Wise Cut Off Marks

NEET Marks vs Rank 2024|How to Calculate?

NEET 2025: All Major Changes in Application Process, Pattern and More
