Find the best tutors and institutes for Class 11 Tuition
Q14.22:
Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.
Solution :
Step 1: Establishing the Equations of Motion
For a particle undergoing Simple Harmonic Motion (SHM) along a straight line, the displacement $x(t)$ as a function of time is given by:
$x(t) = A \sin(\omega t + \phi)$
[Per the definition of SHM, where $A$ is amplitude, $\omega$ is angular frequency, and $\phi$ is the phase constant].
The instantaneous velocity $v(t)$ is the first derivative of displacement with respect to time:
$v(t) = \frac{dx}{dt} = A\omega \cos(\omega t + \phi)$
Step 2: Defining Kinetic and Potential Energy
The instantaneous Kinetic Energy ($K$) is defined as:
$K = \frac{1}{2}mv^2 = \frac{1}{2}m A^2 \omega^2 \cos^2(\omega t + \phi)$
[Using the relation $k = m\omega^2$, where $k$ is the force constant, we can write $K = \frac{1}{2} k A^2 \cos^2(\omega t + \phi)$].
The instantaneous Potential Energy ($U$) is defined as:
$U = \frac{1}{2}kx^2 = \frac{1}{2} k A^2 \sin^2(\omega t + \phi)$
Step 3: Calculating Average Energy Over One Period ($T$)
The average value of a function $f(t)$ over a period $T$ is given by $\langle f \rangle = \frac{1}{T} \int_{0}^{T} f(t) dt$.
For Kinetic Energy:
$\langle K \rangle = \frac{1}{T} \int_{0}^{T} \frac{1}{2} k A^2 \cos^2(\omega t + \phi) dt$
Using the identity $\cos^2\theta = \frac{1 + \cos(2\theta)}{2}$:
$\langle K \rangle = \frac{kA^2}{2T} \int_{0}^{T} \frac{1 + \cos(2(\omega t + \phi))}{2} dt$
Since the integral of $\cos(2\omega t + 2\phi)$ over a full period $T = \frac{2\pi}{\omega}$ is zero:
$\langle K \rangle = \frac{kA^2}{4T} [t]_{0}^{T} = \frac{kA^2}{4T} \cdot T = \frac{1}{4}kA^2$
Step 4: Calculation for Potential Energy
Similarly, for Potential Energy:
$\langle U \rangle = \frac{1}{T} \int_{0}^{T} \frac{1}{2} k A^2 \sin^2(\omega t + \phi) dt$
Using the identity $\sin^2\theta = \frac{1 - \cos(2\theta)}{2}$:
$\langle U \rangle = \frac{kA^2}{2T} \int_{0}^{T} \frac{1 - \cos(2(\omega t + \phi))}{2} dt$
$\langle U \rangle = \frac{kA^2}{4T} [t]_{0}^{T} = \frac{1}{4}kA^2$
Visual Representation of Energy Distribution
Conclusion
Comparing the derived averages:
| Energy Type | Average Value |
|---|---|
| Kinetic Energy ($\langle K \rangle$) | $\frac{1}{4}kA^2$ |
| Potential Energy ($\langle U \rangle$) | $\frac{1}{4}kA^2$ |
Final Solution: Since $\langle K \rangle = \frac{1}{4}kA^2$ and $\langle U \rangle = \frac{1}{4}kA^2$, it is proven that $\langle K \rangle = \langle U \rangle$ for a particle in linear SHM.
More Questions from Class 11 Physics Oscillations Additional Exercises
- Q14.20: An air chamber of volume $V$ has a neck area of cross section $a$ into which a ball of mass $m$ just fits and can move up and down without any friction (Fig.14.27). Show that when the ball is pressed down a little and released , it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal [see Fig. 14.27].
- Q14.21(a): You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (a) the spring constant $k$ and
- Q14.21(b): You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (b) the damping constant $b$ for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.
- Q14.23: A circular disc of mass 10 kg is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be 1.5 s. The radius of the disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional spring constant $\alpha$ is defined by the relation $J = –\alpha \theta$, where $J$ is the restoring couple and $\theta$ the angle of twist).
- Q14.24(a): A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (a) 5 cm
- Q14.24(b): A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (b) 3 cm
- Q14.24(c): A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (c) 0 cm.
- Q14.25: A mass attached to a spring is free to oscillate, with angular velocity $\omega$, in a horizontal plane without friction or damping. It is pulled to a distance $x_0$ and pushed towards the centre with a velocity $v_0$ at time $t = 0$. Determine the amplitude of the resulting oscillations in terms of the parameters $\omega$, $x_0$ and $v_0$. [Hint : Start with the equation $x = a cos (\omega t+\theta)$ and note that the initial velocity is negative.]
CBSE Solutions for Class 11 Physics Oscillations
Chapters in CBSE - Class 11 Physics
Top Tutors who teach Oscillations
Top Tutor
A B Road, Indore
4 yrs of
Exp
1 student
Online Classes
Student's Home
I am a professional tutor with four years of experience in teaching. Having a degree in M.Com from Vikram University and an MBA from IMD Indore I have also completed my B.Ed from IPS Academy Indore in the year 2015. I teach Spoken English, Accountancy, Business Studies, Economics, Statistics, Secretarial Practices, etc., for State, CBSE, ISC/ICSE Board students. I am well versed in English, Hindi and can take classes at Student's Home, My Home and Online Classes via video chat, skype, google hangout, etc.
Top Tutor
Madhurawada, Visakhapatnam
20 yrs of
Exp
47 students
5175 hrs of classes completed
400
per hour
Online Classes
Student's Home
Tutor's home
I mentored several students for CBSE/ICSE/STATE/NIOS 11 th class in physics for the last 24 years.
I recently started taking online physics classes with him, and my experience has been really positive. He is a knowledgeable and dedicated teacher who explains even difficult concepts in a simple and understandable manner. What I appreciate most is his patience and willingness to clear every doubt, ensuring that I have a proper understanding of the topic before moving forward. His teaching approach makes physics much easier to follow and helps build confidence in the subject. He is also very approachable and creates a comfortable learning environment where asking questions never feels difficult. I would definitely recommend him to students looking for a supportive and effective physics tutor.
Top Tutor
Rajajinagar Mahaganapathi Nagar, Bangalore
15 yrs of
Exp
52 students
5141 hrs of classes completed
400
per hour
Online Classes
Tutor's home
I have done Post Graduation in Mathematics, Presidency college, Chennai University. I was a topper and a gold medalist in MSc Math Presidency College. I have been teaching math from last 15 years. I have taken tuitions for 11th Maths for students across different boards like CBSE, ICSE, IGCSE, ISC and State Board. I have an expertise in Math with an excellent track record. My focus is students' success. Conceptually , I make students strong so that they can handle any question in the exam. I give challenging problems in the class so that you can become competetive. Your future is very promising with me.
Mala ma'am makes me understand concepts with clarity and clears all my doubts with lots of patience making to solve problems with ease. Thank you ma'am for always being there to help and support me.
Top Tutor
Behala, Kolkata
6 yrs of
Exp
4 students
1162 hrs of classes completed
1000
per hour
Online Classes
Student's Home
Tutor's home
As a passionate mathematics teacher, I have always enjoyed imparting knowledge to others. I am a B.Tech graduate in Electronics and Communication. I have been teaching mathematics for over four years, and my love for the subject has grown stronger with time. I started teaching as a part-time tutor while pursuing my engineering degree and fell in love with the profession. After completing my degree, I worked with Byjus, one of the leading Edtech companies in India, as a mathematics teacher for two years. While I enjoyed my time at Byju's, I could do more as a teacher and wanted to explore new avenues. I joined Vedantu, an online learning platform, where I have been teaching mathematics for the last two years. I started my journey as a math teacher out of pure passion. Mathematics has always been my favourite subject, and I enjoyed solving problems and equations from a young age. As I grew older, I realised I had a natural talent for teaching, and I decided to pursue it as a career. Currently, I provide mathematics classes for students from sixth to twelfth for all boards, including ICSE, CBSE, and West Bengal. I teach offline and online classes, each lasting for one and a half hours. I use interactive tools such as Zoom, Pentab, and One Note to make my lessons engaging and interactive for my students. To help my students excel in mathematics, I provide them with reference books, notes, and in-built assignments to test their knowledge. I also conduct monthly tests to assess their progress and identify improvement areas. Teaching on UrbanPro has allowed me to interact with students from different parts of the country and help them realise their full potential in mathematics. I want to continue making a positive impact in the lives of my students and help them develop a love for mathematics, just like I did
Sirs class is very interactive, explains each topic in detail making it crystal clear for the student to understand and learn. Always approachable and is ready to clear doubts anytime.
Top Tutor
Garia Baishnabghata Patuli Township, Kolkata
8 yrs of
Exp
1 student
12 hrs of classes completed
400
per hour
Online Classes
Student's Home
Tutor's home
I have 7 years experience in various schools and institutions ( Dips, Rock vale academy, Grand gems public schools) . I can teach each and every topics very easily and make it stronger for the students.
Sayan sir is awesome. He thought us not like a teacher but just like a friend. I am grateful to get such a teacher like Sayan sir.
