<< Chapter < Page Chapter >> Page >
Sketch of the profile of Great Blue Hill, Milton, MA. The summit is 195 meters above sea level. The base of the hill is 147 meters below the summit.
Sketch of the profile of Great Blue Hill, Milton, MA. The altitudes of the three levels are indicated.


First, we need to pick an origin for the y -axis and then determine the value of the constant that makes the potential energy zero at the height of the base. Then, we can determine the potential energies from [link] , based on the relationship between the zero potential energy height and the height at which the hiker is located.


  1. Let’s choose the origin for the y -axis at base height, where we also want the zero of potential energy to be. This choice makes the constant equal to zero and
    U ( base ) = U ( 0 ) = 0 .
  2. At the summit, y = 147 m , so
    U ( summit ) = U ( 147 m ) = m g h = ( 75 × 9.8 N ) ( 147 m ) = 108 kJ .
  3. At sea level, y = ( 147 195 ) m = −48 m , so
    U ( sea-level ) = ( 75 × 9.8 N ) ( −48 m ) = −35.3 kJ .


Besides illustrating the use of [link] and [link] , the values of gravitational potential energy we found are reasonable. The gravitational potential energy is higher at the summit than at the base, and lower at sea level than at the base. Gravity does work on you on your way up, too! It does negative work and not quite as much (in magnitude), as your muscles do. But it certainly does work. Similarly, your muscles do work on your way down, as negative work. The numerical values of the potential energies depend on the choice of zero of potential energy, but the physically meaningful differences of potential energy do not. [Note that since [link] is a difference, the numerical values do not depend on the origin of coordinates.]

Check Your Understanding What are the values of the gravitational potential energy of the hiker at the base, summit, and sea level, with respect to a sea-level zero of potential energy?

35.3 kJ, 143 kJ, 0

Got questions? Get instant answers now!

Elastic potential energy

In Work , we saw that the work done by a perfectly elastic spring, in one dimension, depends only on the spring constant and the squares of the displacements from the unstretched position, as given in [link] . This work involves only the properties of a Hooke’s law interaction and not the properties of real springs and whatever objects are attached to them. Therefore, we can define the difference of elastic potential energy for a spring force as the negative of the work done by the spring force in this equation, before we consider systems that embody this type of force. Thus,

Δ U = W A B = 1 2 k ( x B 2 x A 2 ) ,

where the object travels from point A to point B . The potential energy function corresponding to this difference is

U ( x ) = 1 2 k x 2 + const .

If the spring force is the only force acting, it is simplest to take the zero of potential energy at x = 0 , when the spring is at its unstretched length. Then, the constant is [link] is zero. (Other choices may be more convenient if other forces are acting.)

Spring potential energy

A system contains a perfectly elastic spring, with an unstretched length of 20 cm and a spring constant of 4 N/cm. (a) How much elastic potential energy does the spring contribute when its length is 23 cm? (b) How much more potential energy does it contribute if its length increases to 26 cm?


When the spring is at its unstretched length, it contributes nothing to the potential energy of the system, so we can use [link] with the constant equal to zero. The value of x is the length minus the unstretched length. When the spring is expanded, the spring’s displacement or difference between its relaxed length and stretched length should be used for the x -value in calculating the potential energy of the spring.


  1. The displacement of the spring is x = 23 cm 20 cm = 3 cm , so the contributed potential energy is U = 1 2 k x 2 = 1 2 ( 4 N/cm ) ( 3 cm ) 2 = 0.18 J .
  2. When the spring’s displacement is x = 26 cm 20 cm = 6 cm , the potential energy is U = 1 2 k x 2 = 1 2 ( 4 N/cm ) ( 6 cm ) 2 = 0.72 J , which is a 0.54-J increase over the amount in part (a).


Calculating the elastic potential energy and potential energy differences from [link] involves solving for the potential energies based on the given lengths of the spring. Since U depends on x 2 , the potential energy for a compression (negative x ) is the same as for an extension of equal magnitude.

Got questions? Get instant answers now!

Questions & Answers

Damping is provided by tuning the turbulence levels in the moving water using baffles.How it happens? Give me a labelled diagram of it.
Shaina Reply
A 10kg ball travelling at 4meter per second collides elastically in a head-on collision with a 2kg ball.What are (a)the velocities and (b)the total momentum of the balls after collision?
Law Reply
a)v1 8/3s&v2 20/3s. b)in elastic collision total momentum is conserved.
The displacement of the air molecules in sound wave is modeled with the wave function s(x,t)=5.00nmcos(91.54m−1x−3.14×104s−1t)s(x,t)=5.00nmcos(91.54m−1x−3.14×104s−1t) . (a) What is the wave speed of the sound wave? (b) What is the maximum speed of the air molecules as they oscillate in simple harmon
Shaina Reply
practical 1st year physics
Nsc Reply
Whats the formular for newton law of motion
Ahmad Reply
F=m×a Where F=force M=mass of a body of an object a=acceleration due to gravity
what is speed
Hassan Reply
distance travelled per unit of time is speed.
distance travelled in a particular direction it is.
Speed is define as the distance move per unit time. Mathematically is given as Speed = distance/time Speed = s/t
speed is a vector quantity. It is defined distance per unit time.It's unit in c.g.s cm/s and in S.I. m/s.It’s dimension is LT^-1
formula for velocity
Amraketa Reply
v=ms^-1 velocity=distance time
(p-a/v)(v-b)=nrt what is the dimension of a
velocity= displacement time
Velocity = speed/time
what are evasive medical diagnosis?
Shaina Reply
If the block is displaced to a position y , the net force becomes Fnet=k(y−y0)−mg=0Fnet=k(y−y0)−mg=0 . But we found that at the equilibrium position, mg=kΔy=ky0−ky1mg=kΔy=ky0−ky1 . Substituting for the weight in the equation yields. Show me an equation of graph.
simple harmonic motion defination
Maharam Reply
how to easily memorize motion equation
how speed destrog is uranium
Sayed Reply
where can we find practice problems?
bonokuhle Reply
I'm not well
can u tell me the expression for radial acceleeation
Shikha Reply
Is equal to the square of the velocity divided by the radius of circular path of the object
how to find maximum acceleration and velocity of simple harmonic motion?
how to find maximum acceleration and velocity of simple harmonic motion and where it occurres?
you can use either motion equations or kinetic equation and potential equation .
how destraction 1kg uranium
A Radial Acceleration is defined as the upward movement of an object.
A body of 2.0kg mass makes an elastic collision with another at rest and continues to more in the original direction but with 1/4 of its ori is the mass of the struck body?
bright Reply
pls help me solve this problem
why do sound travel faster in the night than in the day
Isaac Reply
I believe because speed is a function of air density, and colder air is more dense
At night air is denser because of humidity.
Night air is cooler. Sound requires medium to travel so the denser the medium the fastest the sound travels. Humid air is denser then warmer air as in day.
The humidity statement is misleading , colder air is more dense period.
because there is no any other sound to reverberate with it so it clearly travel to lot of distance and also humidity and also due to denser air at night
please could you guys help me with physics best websites
because it is quiet at night. this takes us to the topic wave, it depends on the wave at that moment, which Echo's....sound travelled.
because it is quite at night. this takes us to the topic wave , it depends on the wave at that the physics
what is an atomic radius
Peace Reply
element radioactivit diffusion atomic radius alpha beta gamma these elements have more than 92 atomic mass start from uranium
Practice Key Terms 2

Get the best University physics vol... course in your pocket!

Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?