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# 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:

F_{T}= F_{g}

F_{g}= mg

where

**F**is the upward force_{T}**F**is the resistive force of gravity_{g}**m**is the mass of the object**g**is the acceleration due to gravity

The work required to lift the object is:

W = mgy

where

**W**is the work required**y**is the displacement of the object

## Accelerating upward

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:

F_{T}= F_{i}+ F

F_{T}= ma + mg

where

**F**is the resistive force of inertia_{i}**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)y_{1}+ mgy_{2}

## 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:

F_{T}= F_{g}+ F_{i}+ F_{a}

F_{T}= mg + ma + kv^{2}

where

**F**is the force of the air resistance or drag_{a}**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:

W_{T}= F_{T}y

W_{T}= (mg + ma + kv^{2})y

## Summary

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

### Websites

**Work by gravity** by Sunil Kumar Singh - Connexions

**Gravity and Inertia in Running** - Locomotion and Biology paper (PDF)

### Books

**Top-rated books on Simple Gravity Science**

**Top-rated books on Advanced Gravity Physics**

## Questions and comments

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## Work Against Gravity to Lift an Object