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Work Done by Gravity Against Inertia and Air Resistance
by Ron Kurtus (updated 29 May 2023)
When an object is falling freely, the force of gravity is doing work against the resistance from inertia and the air resistance or drag on the object.
The forces acting on the object are gravity and the opposite or resistive forces of inertia and air resistance. When the object is moving slowly, air resistance is negliable, and the resistance is only due to inertia from the acceleration of the object. At some velocity, air resistance is equal to the force of gravity, and the object no longer accelerates. This is called the terminal velocity of the object.
The work done equals the product of the force of gravity and the displacement of the object. It can also be determined by the change in potential energy of the object due to gravity.
Questions you may have include:
- What are the forces on a falling object?
- What is the work related to displacement?
- How is work determined from potential energy?
This lesson will answer those questions. Useful tool: Units Conversion
Forces on a falling object
The force of gravity pulls objects toward the Earth. The resistance to the pull of gravity consists of inertia from the object's acceleration and air resistance from the object's velocity.
Total force
According to Newton's Law of Action-Reaction, the force of gravity equals the resistive forces for a freely falling object.
F_{g} = F_{i} + F_{a}
where
- F_{g} is the force of gravity
- F_{i} is the resistance from inertia
- F_{a} is the air resistance force
Force of gravity
The force of gravity to accelerate an an object is constant:
F_{g} = mg
where
- F_{g} is the force of gravity in newtons (N) or pound-force (lbs)
- m is the mass of the object in kilograms (kg) or pound-mass (lbs)
- g is the acceleration due to gravity (9.8 m/s^{2} or 32 ft/s^{2})
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-mass equals 32 pound-force.
Resistance from inertia
As an object accelerates during a free fall, the resistance of inertia increases, according to Newton's Law of Inertia. The resistive force of inertia is:
F_{i} = ma
where
- F_{i} is the force of inertia resisting acceleration
- a is the rate of acceleration
Air resistance or drag
The air resistance force or drag is:
F_{a} = kv^{2}
where
- F_{a} is the air resistance or drag force
- k is a constant dependent on density and shape of the object
- v is the velocity of the object
Negligible air resistance
For large masses or at low velocities, air resistance can be considered negligible. This is the usual assumption in equations for falling objects. In such a case:
F_{g} = F_{i}
and
mg = ma
For example, the experiment of dropping an object in the lab or even dropping two lead balls from the Leaning Tower of Pisa, the effect of air resistance can be ignored.
Terminal velocity
However, at some velocity, air resistance can equal the force of gravity, resulting in zero resistance from inertia.
kv^{2} = mg = F_{g}
F_{g} = F_{i} + F_{g}
F_{i} = 0
No acceleration means the velocity is constant.
For example, when dropping a coin from a tall building, the air resistance will cause the coin to reach a terminal velocity, when it no longer accelerates while falling.
In either case, the force of gravity—and thus the work done by gravity—is the same.
Work as force times displacement
The general equation for work is:
W = Fy
where
- W is the work done against inertia in joules (J) or pound-feet
- F is the force applied to an object in newtons (N) or pound-force (lbs)
- y is the displacement of the object while the force is applied in meters (m) or feet (ft)
Note: You may often see the word distance used in work. To be scientifically correct, displacement should be used instead. Distance can follow any path, while displacement is a vector and straight path in the line of the force.
(See Convention for Direction in Gravity Equations for more information.)
Work done by gravity
The work done by gravity to overcome resistance from inertia and air resistance is:
W = (F_{i} + F_{a})y
W = F_{g}y
W = mgy
where
- W is the work done in joules (J) or pound-feet
- y is the vertical displacement in m or ft from the starting point to some end point
Work by gravity as a function of displacement
Work as change in potential energy
The amount of work done by gravity to overcome the resistance of inertia can also be defined as the change in the potential energy.
Proof of that relationship starts with the equation for the potential energy of an object due to the force of gravity:
PE = mgh
where
- PE is the potential energy in joules (J) or foot-pounds (ft-lbs)
- h is the height above the ground in m or ft
(See Potential Energy of Gravity for more information.)
The change in potential energy is:
ΔPE = mgh_{i} − mgh_{f}
where
- Δ is the Greek letter delta, indicating a change or difference
- h_{i} is the initial height above the ground
- h_{f} is the final height above the ground
Derive equation for work
Let y be the displacement the object falls from the starting point above the ground:
y = h_{i} − h_{f}
Multiplying both sides of equation by mg:
mgy = mgh_{i} − mgh_{f}
Thus:
mgy = ΔPE
W = ΔPE = mgy
An illustration of this is:
Work as change in potential energy
Summary
The forces acting on a freely falling object are gravity and the resistive forces of inertia and air resistance. When the object is moving slowly, air resistance is neg liable. At the terminal velocity of the falling object, air resistance is equal to the force of gravity, and the object no longer accelerates.
The work done equals the product of the force of gravity and the displacement of the object. It can also be determined by the change in potential energy of the object due to gravity.
Be conscientious
Resources and references
Websites
Work by gravity by Sunil Kumar Singh - Connexions
Gravity and Inertia in Running - Locomotion and Biology paper (PDF)
Forces on a Falling Object in Air - NASA
Drag - Wikipedia
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Work Done by Gravity Against Inertia and Air Resistance