Work Against Gravity to Lift an Object
by Ron Kurtus (revised 27 June 2013)
When an object is lifted or projected upward, work must be done against the resistance from gravity. In some situations, the resistance of inertia from accelerating the object and air resistance must be taken into account.
If the object is already moving upward at some initial velocity, the work done by gravity is simply the force of gravity times the displacement, provided the velocity is constant and small enough that air resistance is neg liable.
If the object is accelerated during the lifting process, the resistance from inertia must be taken into account. If the object is projected upward at a high velocity, air resistance must be added to the equation.
Questions you may have include:
- What is the work done when the object is lifted at a constant velocity?
- What is the work when the object is accelerated upward?
- What happens when the object is projected upward at a high velocity?
This lesson will answer those questions. Useful tool: Units Conversion
Already moving upward
If an object has an initial upward velocity and some force is continuing to move it upward at that constant velocity, the force required to move the object equals the restive force of gravity:
FT = Fg
Fg = mg
- FTis the upward force
- Fg is the resistive force of gravity
- m is the mass of the object
- g is the acceleration due to gravity
The work required to lift the object is:
W = mgy
- W is the work required
- y is the displacement of the object
If an object is stationary or moving upward at some initial velocity, a sufficient upward force can accelerate the object to higher velocities, The upward force must be greater than the downward force of gravity in order to accelerate the object upward.
Most often the air resistance in relatively low upward acceleration is negligible.
The upward force is:
FT = Fi + F
FT = ma + mg
- Fi is the resistive force of inertia
- a is the acceleration of the object
Th resulting work required to lift and accelerate an object is:
W = m(a + g)y
If the object is accelerated to some height and then is continued to be lifted at a constant velocity, the work becomes:
W = m(a + g)y1 + mgy2
High velocity acceleration
When an object, such as a rocket, is acceleration upwards at a high velocity, air resistance becomes an added factor in the work required.
The total upward force to propel a rocket upward is:
FT = Fg + Fi + Fa
FT = mg + ma + kv2
- Fa is the force of the air resistance or drag
- k is the air resistance constant related to the size and shape of the object
- v is the velocity of the object
The work required to propel a rocket to some altitude is:
WT = FTy
WT = (mg + ma + kv2)y
When an object is lifted or projected upward, work must be done against the resistance from gravity, inertia and air resistance.
If the object is moving upward at constant velocity, the work done by gravity is simply the force of gravity times the displacement. If the object is accelerated during the lifting process, the resistance from inertia must be taken into account. If the object is projected upward at a high velocity, air resistance must be added to the equation.
Hard work will get you far in life
Resources and references
Work by gravity by Sunil Kumar Singh - Connexions
Gravity and Inertia in Running - Locomotion and Biology paper (PDF)
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