•  
agra,ahmedabad,ajmer,akola,aligarh,ambala,amravati,amritsar,aurangabad,ayodhya,bangalore,bareilly,bathinda,bhagalpur,bhilai,bhiwani,bhopal,bhubaneswar,bikaner,bilaspur,bokaro,chandigarh,chennai,coimbatore,cuttack,dehradun,delhi ncr,dhanbad,dibrugarh,durgapur,faridabad,ferozpur,gandhinagar,gaya,ghaziabad,goa,gorakhpur,greater noida,gurugram,guwahati,gwalior,haldwani,haridwar,hisar,hyderabad,indore,jabalpur,jaipur,jalandhar,jammu,jamshedpur,jhansi,jodhpur,jorhat,kaithal,kanpur,karimnagar,karnal,kashipur,khammam,kharagpur,kochi,kolhapur,kolkata,kota,kottayam,kozhikode,kurnool,kurukshetra,latur,lucknow,ludhiana,madurai,mangaluru,mathura,meerut,moradabad,mumbai,muzaffarpur,mysore,nagpur,nanded,narnaul,nashik,nellore,noida,palwal,panchkula,panipat,pathankot,patiala,patna,prayagraj,puducherry,pune,raipur,rajahmundry,ranchi,rewa,rewari,rohtak,rudrapur,saharanpur,salem,secunderabad,silchar,siliguri,sirsa,solapur,sri-ganganagar,srinagar,surat,thrissur,tinsukia,tiruchirapalli,tirupati,trivandrum,udaipur,udhampur,ujjain,vadodara,vapi,varanasi,vellore,vijayawada,visakhapatnam,warangal,yamuna-nagar

Potential Energy: Definition, Formula, Derivation of Gravitational Potential Energy

Potential Energy: Definition, Formula, Derivation of Gravitational Potential Energy

Have you ever wondered what energy does water stored in a tank possess when it is placed at a height h above the surface of the earth? Or a car parked on a hilltop several kilometers above the ground? Is there energy at all in the first place? Yes there is energy, and it is evident from the fact that when you let the hand brake of the car free or allow the water to come out, you can see them gaining kinetic energy. From where did this kinetic energy come? It came at the cost of potential energy.

The higher an object is located above the surface of the earth, the higher is its potential energy. In the following diagram for instance, the ball m, when located at the top of a tower of height h has a higher potential energy than when it was located at the bottom of the tower. Another way to imagine potential energy would be this: imagine you lifted a bottle of water from the ground up to your desk. The work done by your hand is stored as potential energy in the bottle. 

 



      Fig. A ball of mass m at two different energy levels

Table of Contents

  • Gravitational Potential Energy Derivation
  • Practice Problems of Potential Energy
  • FAQs of Potential Energy

Definition of Potential Energy

Potential energy (P.E) can be defined as the energy possessed by a body by virtue of its position in a conservative force field; i.e a field in which work done by the forces do not depend upon the path taken in moving the body. If a conservative force F with arrow acting on a body displaces it through a distance d barr, then the change in P.E. can be written as 

U- U∫ F with arrow.d barr; here Ui and Uf indicate the initial and final potential energy. 

Note: Change in potential energy is △U = U- Ui=-W; where W = ∫ F with arrow.d barr is the work done by the conservative force.

Gravitational Potential Energy Derivation

Let us assume a ball of mass m is moved to a height h vertically. The work done in shifting the ball through a height is stored as gravitational potential energy inside the ball. 

Change in P.E in this process for a small displacement dy can be written as mgdy

pic

Taking the ground as reference level as the potential energy datum, where the P.E is zero; i.e Ui=0, the above equation becomes

 U- U= U- 0 = mgh


 

Practice Problems 

Q1) Calculate the potential energy stored in a bowl of mass 100 g when it is raised through a height of 0.2 m. (Take g=10 ms-2)

Solution) Given, m=0.1 kg

h=0.2 m

Work done = potential energy stored in the bowl = mgh = 0.1 × 10 × 0.2 = 0.2 J

Q2) A chain of mass M and length l hangs on a frictionless table. 23rd of the chain lies on the table. The work done in pulling the entire chain up the table will be?


pic2

Q3) A uniform rod of mass M and length l is held in position as shown in the figure. Find the potential energy of the rod.


Solution) 


Given, mass of the rod = M

Length of the rod = l

Consider an element of length dx and mass dm at a distance x from the bottom of the rod. Then 

pic3

Q4) Find the gravitational potential energy of the chain with reference level at the center of the hemisphere. 


Solution) Let dm be the mass of an element of length dx and m be the mass of the chain. 


pic4

FAQs 

Question1. Can potential energy be defined in non-conservative field?
Answer.
Potential energy can only be defined in conservative fields.

Question2. What is the expression for gravitational potential of a particle of mass m raised through a height h?(-acceleration due to gravity)
Answer.
mgh

Question3. Is potential energy a scalar or a vector quantity?
Answer.
P.E is a scalar quantity.

Question4. What is the dimensional formula for potential energy?
Answer.
 [ML2T-2].

Aakashians JEE Topper

Aakashians NEET UG 2023 Champions Again

Historic Results 2023 Aakashians Qualified +1 Lakh

JEE Advanced 2023

JEE Advanced 2023 Stats

JEE Advanced 2022 Topper

NEET Related Links

NEET Exam 2024

NEET 2024 Exam Dates

NEET 2024 Exam pattern

NEET 2024 Syllabus

NEET 2024 Eligibility Criteria

NEET 2024 Application

NEET UG Counselling

NEET FAQ

NEET UG Result

NEET 2024 Cut Off

Neet 2023 Toppers List Names & Rank

Neet Result 2023 Toppers list rank cut off

Neet Answer key Live Download PDF

Neet 2023 State Toppers List

JEE MAIN Related Links

JEE Main 2024

JEE Main Rank Predictor 2024

JEE Main College Predictor 2024

JEE Main 2024 Exam Dates

JEE Main 2024 Exam pattern

JEE Main 2024 Application

JEE Main 2024 Eligibility Criteria

JEE Main 2024 Syllabus

JEE Main 2024 Physics Syllabus

JEE Main 2024 Maths Syllabus

JEE Main 2024 Chemistry Syllabus

JEE Main 2024 Admit Card

JEE Main 2024 Counselling

JEE Main marks vs rank vs percentile

JEE Advanced Result 2023 live topper list

JEE Exam Preparation - How to calculate your rank jee

JEE Maths Syllabus - Important topics and weightage

JEE Advanced Related Links

JEE Advanced 2024 Exam Dates

JEE Advanced 2024 Application

JEE Advanced 2024 Eligibility Criteria

JEE Advanced 2024 Syllabus

JEE Advanced 2024 Maths Syllabus

JEE Advanced 2024 Physics Syllabus

JEE Advanced 2024 Chemistry Syllabus

JEE Advanced Exam Result

JEE Advanced Exam Dates

JEE Advanced Registration Dates

CUET Related Links

CUET 2024 Eligibility Criteria

CUET 2024 Admit Card

CUET 2024 Exam Pattern

CUET 2024 FAQs

CUET 2024 Counselling

CUET 2024 Syllabus

CUET 2024 Result

CUET 2024 Answer Key

CUET 2024 Preparation

CUET CUTOFF

CUET 2024 Application Form

Talk to our expert
Resend OTP Timer =
By submitting up, I agree to receive all the Whatsapp communication on my registered number and Aakash terms and conditions and privacy policy