The other force on the lawn mower mentioned above was Earth’s gravitational force, or the weight of the mower. Near the surface of Earth, the gravitational force on an object of mass
m has a constant magnitude,
mg , and constant direction, vertically down. Therefore, the work done by gravity on an object is the dot product of its weight and its displacement. In many cases, it is convenient to express the dot product for gravitational work in terms of the
x -,
y -, and
z -components of the vectors. A typical coordinate system has the
x -axis horizontal and the
y -axis vertically up. Then the gravitational force is
so the work done by gravity, over any path from
A to
B , is
The work done by a constant force of gravity on an object depends only on the object’s weight and the difference in height through which the object is displaced. Gravity does negative work on an object that moves upward (
), or, in other words, you must do positive work against gravity to lift an object upward. Alternately, gravity does positive work on an object that moves downward (
), or you do negative work against gravity to “lift” an object downward, controlling its descent so it doesn’t drop to the ground. (“Lift” is used as opposed to “drop”.)
Shelving a book
You lift an oversized library book, weighing 20 N, 1 m vertically down from a shelf, and carry it 3 m horizontally to a table (
[link] ). How much work does gravity do on the book? (b) When you’re finished, you move the book in a straight line back to its original place on the shelf. What was the total work done against gravity, moving the book away from its original position on the shelf and back again?
Side view of the paths for moving a book to and from a shelf.
Strategy
We have just seen that the work done by a constant force of gravity depends only on the weight of the object moved and the difference in height for the path taken,
. We can evaluate the difference in height to answer (a) and (b).
Solution
Since the book starts on the shelf and is lifted down
, we have
There is zero difference in height for any path that begins and ends at the same place on the shelf, so
Significance
Gravity does positive work (20 J) when the book moves down from the shelf. The gravitational force between two objects is an attractive force, which does positive work when the objects get closer together. Gravity does zero work (0 J) when the book moves horizontally from the shelf to the table and negative work (−20 J) when the book moves from the table back to the shelf. The total work done by gravity is zero
Unlike friction or other dissipative forces, described in
[link] , the total work done against gravity, over any closed path, is zero. Positive work is done against gravity on the upward parts of a closed path, but an equal amount of negative work is done against gravity on the downward parts. In other words, work done
against gravity, lifting an object
up , is “given back” when the object comes back down. Forces like gravity (those that do zero work over any closed path) are classified as conservative forces and play an important role in physics.
