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Work by Gravity Against Inertia
by Ron Kurtus (23 December 2009)
Work is defined as a force acting on an object and moving it a distance against some resistance. This work also results in a change of mechanical energy. Inertia is the resistance to changing the velocity or direction of motion of an object.
The force of gravity acting on an object to overcome the resistance from inertia is work done by gravity. This applies whether the object is stationary, moving upward or moving downward.
Questions you may have include:
- What is the work against inertia when the object is initially stationary?
- What is the work when the object is moving downward?
- What is the work done when the object is moving upward?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
Object initially stationary
When you hold and object and drop it from some height, the force of gravity accelerates the object, overcoming the resistance of inertia. This results in work being done, as seen by a change in the object's kinetic energy.
Work to overcome inertia
The work done to overcome inertia is:
W = Fd
where
- W is the work in newton-meters (N-m) or foot-pound-force (ft-lb)
- F is the applied force in newtons (N) or pound-force (lbs)
- d is the distance the object is moved in meters (m) or feet (ft)
Work done by gravity to overcome inertia
For an object relatively close to Earth, the force of gravity applied to the object is:
F = mg
where
- m is the mass of the object in kilograms (kg) or pound-mass (lbs)
- g is the acceleration due to gravity (9.8 m/s2 or 32 ft/s2)
Note: Pounds are typically considered units of force or weight. However, some people also use the expression "pound" when referring to mass. Thus, the unit of pound-force is used to distinguish it from pound-mass. Also, since F = mg, 1 pound-force = 32 pound-mass.
Since neither m nor g change, the force is a constant value, and the work done by gravity is simply:
W = mgy
where y is the distance from the starting to end point in the direction toward the Earth.
Work by gravity against inertia
Work is change in energy
Work also results in a change of kinetic energy. At the initial position, the object is not moving, so its kinetic energy is zero. Determining the final kinetic energy provides the change.
From the lesson Gravity Distance Equations for Falling Objects, the relationship between the distance the object has fallen and its velocity at that point is:
y = v2/2g
Thus:
W = mgv2/2g
W = mv2/2
This is the kinetic energy at the final position and is also the difference between the initial position and the end position.
Object moving downward
If an object is initially moving downward at some velocity, the force of gravity will accelerate the object, doing work against inertia, according to:
W = mgy
The work done is independent of the initial velocity. However, that velocity is a factor in the change in kinetic energy.
From the lesson Gravity Distance Equations for Objects Projected Downward, the distance for a given distance from the starting point is:
y = (vf2− vi2)/2g
where
- vi is the initial velocity
- vf is the final velocity
Thus the work done is:
W = mvf2/2 − mvi2/2
Note: If vi = 0, this example reduces to the case where the object is initially stationary.
Since kinetic energy equals mv2/2, the work is equal to the change in kinetic energy. This is often written as:
W = ΔKE
where
- Δ is the Greek capital letter delta, meaning "increment of change"
- KE is kinetic energy
Object moving upward
If an object is initially moving upward at some velocity, the force of gravity will slow the object down until it stops and reverses direction.
Work done by gravity
The work done by gravity in slowing down the object on its way up is:
W = mgy
However, because of our convention that y is negative in the upward direction, the value of W will also be negative. This follows that gravity is not doing work on the object but is taking work away from it.
Work is change in energy
From the lesson Gravity Distance Equations for Objects Projected Upward, the distance for a given distance from the starting point is:
y = (vf2 − vi2)/2g
Thus the work done is:
W = mvf2/2 − mvi2/2
Since the final velocity is less than the initial velocity, the work has a negative value. Also, the work is equal to the change in kinetic energy.
Work and energy on the way down
The object will slow down as it goes upward until it reaches the maximum height, where the velocity equals zero and the work equals:
W = −mvi2/2
As the object accelerates downward, the work is again:
W = mvf2/2 − mvi2/2
When the object reaches the starting point, y = 0, the work done by gravity is zero. Another way to look at it is that the total work is:
Wtotal = Wup − Wdown
Summary
The force of gravity overcomes the resistance of inertia to perform work on a falling object, whether its initial velocity is zero or some downward value. If the object has an upward initial velocity, the work by gravity to overcome inertia is a negative value. Work results in a change in kinetic energy of the object.
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Resources
The following resources provide information on this subject:
Websites
Acceleration due to Gravity Calculations - from Western Washington University
Gravity and Gravitation Resources
Books
Top-rated
books on Simple Gravity Science
Top-rated
books on Advanced Gravity Physics
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
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