1. Home
  2. College physics
  3. Work, energy, and energy resources
  4. Gravitational potential energy

7.3 Gravitational potential energy  (Page 2/5)

<< Chapter < Page
  College physics     Page 2 / 5
Chapter >> Page >

More precisely, we define the change in gravitational potential energy Δ PE g size 12{Δ"PE" rSub { size 8{g} } } {} to be

Δ PE g = mgh , size 12{Δ"PE" rSub { size 8{g} } = ital "mgh"} {}

where, for simplicity, we denote the change in height by h size 12{h} {} rather than the usual Δ h size 12{Δh} {} . Note that h size 12{h} {} is positive when the final height is greater than the initial height, and vice versa. For example, if a 0.500-kg mass hung from a cuckoo clock is raised 1.00 m, then its change in gravitational potential energy is

mgh = 0.500 kg 9.80 m/s 2 1.00 m = 4.90 kg ⋅ m 2 /s 2 = 4.90 J.

Note that the units of gravitational potential energy turn out to be joules, the same as for work and other forms of energy. As the clock runs, the mass is lowered. We can think of the mass as gradually giving up its 4.90 J of gravitational potential energy, without directly considering the force of gravity that does the work .

Using potential energy to simplify calculations

The equation Δ PE g = mgh size 12{Δ"PE" rSub { size 8{g} } = ital "mgh"} {} applies for any path that has a change in height of h size 12{h} {} , not just when the mass is lifted straight up. (See [link] .) It is much easier to calculate mgh size 12{ ital "mgh"} {} (a simple multiplication) than it is to calculate the work done along a complicated path. The idea of gravitational potential energy has the double advantage that it is very broadly applicable and it makes calculations easier. From now on, we will consider that any change in vertical position h size 12{h} {} of a mass m size 12{m} {} is accompanied by a change in gravitational potential energy mgh size 12{ ital "mgh"} {} , and we will avoid the equivalent but more difficult task of calculating work done by or against the gravitational force.

There is a four-story building. A person is carrying a television up the stairs of the building. The height of third story is h from the ground. A girl is standing outside the building and is lifting a similar television with the help of a pulley.
The change in gravitational potential energy ( Δ PE g ) size 12{ \( Δ"PE" rSub { size 8{g} } \) } {} between points A and B is independent of the path. Δ PE g = mgh size 12{Δ"PE" rSub { size 8{g} } = ital "mgh"} {} for any path between the two points. Gravity is one of a small class of forces where the work done by or against the force depends only on the starting and ending points, not on the path between them.

The force to stop falling

A 60.0-kg person jumps onto the floor from a height of 3.00 m. If he lands stiffly (with his knee joints compressing by 0.500 cm), calculate the force on the knee joints.

Strategy

This person’s energy is brought to zero in this situation by the work done on him by the floor as he stops. The initial PE g size 12{"PE" rSub { size 8{g} } } {} is transformed into KE size 12{"KE"} {} as he falls. The work done by the floor reduces this kinetic energy to zero.

Solution

The work done on the person by the floor as he stops is given by

W = Fd cos θ = − Fd , size 12{W= ital "Fd""cos"θ= - ital "Fd"} {}

with a minus sign because the displacement while stopping and the force from floor are in opposite directions ( cos θ = cos 180º = − 1 ) size 12{ \( "cos"θ="cos""180""°=" - 1 \) } {} . The floor removes energy from the system, so it does negative work.

The kinetic energy the person has upon reaching the floor is the amount of potential energy lost by falling through height h size 12{h} {} :

KE = − Δ PE g = − mgh , size 12{"KE"= - Δ"PE" rSub { size 8{g} } = - ital "mgh"} {}

The distance d size 12{d} {} that the person’s knees bend is much smaller than the height h size 12{h} {} of the fall, so the additional change in gravitational potential energy during the knee bend is ignored.

The work W size 12{W} {} done by the floor on the person stops the person and brings the person’s kinetic energy to zero:

W = − KE = mgh . size 12{W= - "KE"= ital "mgh"} {}

Combining this equation with the expression for W size 12{W} {} gives

− Fd = mgh . size 12{ - ital "Fd"= ital "mgh"} {}

Recalling that h size 12{h} {} is negative because the person fell down , the force on the knee joints is given by

F = − mgh d = − 60.0 kg 9.80 m /s 2 − 3 . 00 m 5 . 00 × 10 − 3 m = 3 . 53 × 10 5 N. size 12{F= - { { ital "mgh"} over {d} } = - { { left ("60" "." 0" kg" right ) left (9 "." "80"" m/s" rSup { size 8{2} } right ) left ( - 3 "." "00"`m right )} over {5 "." "00" times "10" rSup { size 8{ - 3} } " m"} } =3 "." "53" times "10" rSup { size 8{5} } `N "." } {}

Discussion

Such a large force (500 times more than the person’s weight) over the short impact time is enough to break bones. A much better way to cushion the shock is by bending the legs or rolling on the ground, increasing the time over which the force acts. A bending motion of 0.5 m this way yields a force 100 times smaller than in the example. A kangaroo's hopping shows this method in action. The kangaroo is the only large animal to use hopping for locomotion, but the shock in hopping is cushioned by the bending of its hind legs in each jump.(See [link] .)

Got questions? Get instant answers now!

Questions & Answers

material that allows electric current to pass through
Deng Reply
material which don't allow electric current is called
Deng
insulators
Covenant
how to study physic and understand
Ewa Reply
what is conservative force with examples
Moses
what is work
Fredrick Reply
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
AI-Robot
why is it from light to gravity
Esther Reply
difference between model and theory
Esther
Is the ship moving at a constant velocity?
Kamogelo Reply
The full note of modern physics
aluet Reply
introduction to applications of nuclear physics
aluet Reply
the explanation is not in full details
Moses Reply
I need more explanation or all about kinematics
Moses
yes
zephaniah
I need more explanation or all about nuclear physics
aluet
Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity
Muhammad Reply
yh
Isaac
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
Majok Reply
what is frontier of physics
Somto Reply
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Gufraan Reply
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
Ezekiel Reply
please explain
Samuel
what's the definition of physics
Mobolaji Reply
what is physics
Nangun Reply
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
AI-Robot
<< Chapter < Page Page > Chapter >>
Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask