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# Work by Gravity Against Inertia

by Ron Kurtus (revised 3 February 2011)

The amount of work is the force required to move an object some displacement against a resistance. It is the product of the force and the displacement caused by that force.

The inertia of matter is one such resistance to motion. When an object is dropped or projected downward, gravity does work to overcome the natural inertia of matter. Thus, the product of the force of gravity and the displacement is the work done by gravity against inertia.

Work is also defined as the change in mechanical energy of the object as it moves from one position to another.

In the case of the force of gravity, work can be measured as the change in the potential energy or the change in the kinetic energy of the object. Equating the change in energies allows you to calculate the final velocity of the object after it moves a given displacement.

When the object is projected upward, the work above the starting point is negative and equals zero when the object returns to the starting point..

Questions you may have include:

• What is the equation for work?
• How is work related to energy?
• What happens when an object is projected upwards?

This lesson will answer those questions.

Useful tool: Metric-English Conversion

## Work as force times displacement

A force is required to overcome the resistance of inertia and accelerate an object. As long as the force is being applied, the object will accelerate and work will be done against inertia.

The product of the displacement of the object and the force applied equals the work done against inertia.

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.

The force of gravity to accelerate an object is:

F = mg

where

• F 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/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-mass equals 32 pound-force.

The work done by gravity to overcome inertia is:

W = Fy

W = mgy

where

• W is the work done in joules (J) or pound-feet
• y is the vertical displacement from the starting point to some end point in m or ft

Although F and d are vector quantities with an indicated direction, W is a scalar quantity, with only magnitude and no direction.

### Work against gravity

The force of gravity resists motion in its opposite direction. If an upward force equal to the force of gravity—or the weight of an object—is applied to a stationary object, the forces equal out, and the object does not move. However, if the object has an initial upward velocity and a force equal to gravity is applied, the object will continue to move upward at that initial velocity.

The upward force is:

−Fg = m(−gu)

Fg = mgu

where

• −Fg is the upward force needed to counter the force of gravity in newtons or pounds-force
• m is the mass of the object in kilograms or pounds-mass
• −gu is the acceleration in the opposite direction of the acceleration due to gravity (−9.8 m/s2 or −32 ft/s2)

Note: According to our convention for direction in gravity equations, Fg and gu are negative numbers, since they are in the opposite direction of gravity.

The work done in moving an object against gravity a certain displacement at an initial upward velocity is:

W = (−Fg)*(−y)

W = Fgy

where

• W is the work done against gravity in joules (J) or foot-pounds-force
• −y is the vertical displacement in meters (m) or feet (ft), measured from the starting point to when the force is discontinued

Note: Our convention states that y is negative when it is in the opposite direction of gravity.

Thus:

W = m(−gu)(−y)

W = mguy

Note: When talking about work against gravity, most physics textbooks use h for height: W = mgh. However, you need to remember that h is the displacement that an object is lifted above the ground, while y is the displacement from some starting point at or above the ground.

### Work against inertia

Inertia is a resistance to changing the motion of an object. Its equation is:

−Fi = m(−au)

Fi = mau

where

• −Fi is the force required to overcome the inertia in newtons (N) or pounds-force (lbs)
• m is the mass of the object in kilograms (kg) or pounds-mass
• −au is the upward acceleration of the object in m/s2 or ft/s2

The work against inertia in accelerating an object a displacement upward is:

W = Fiy

W = mauy

Note: At this point, we are not considering accelerating the object against the force of gravity. This equation is for the general work against inertia.

where

• W is the work done in joules (J) or pound-feet
• y is the vertical displacement from the starting point to some end point in m or ft

Work by gravity against inertia

## Work as change in energy

The amount of work done by gravity to overcome the resistance of inertia can also be defined as either the change in potential energy (PE) or as the change in kinetic energy (KE) over the displacement:

W = ΔPE

and

W = ΔKE

where Δ is the Greek letter delta, indicating a change or difference.

### Work as change in potential energy

The equation for potential energy of gravity is:

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

The change in potential energy is:

ΔPE = mghi − mghf

where

• hi is the initial height from the ground
• hf is the final height from the ground

Since y is the displacement the object falls from the starting point above the ground:

y = hi − hf

Multiplying both sides of equation by mg:

mgy = mghi − mghf

Thus:

mgy = ΔPE

W = ΔPE = mgy

An illustration of this is:

Work as change in potential energy

### Work as change in kinetic energy

The equation for kinetic energy is:

KE = mv2/2

The change in kinetic energy is:

ΔKE = mvf2/2 − mvi2/2 = W

where

• vf is the final velocity
• vi is the initial velocity

An illustration of this is:

Work as change in kinetic energy

### Velocity for given displacement

Since ΔPE = ΔKE, you can find the final velocity for work done against inertia moving a given displacement:

mgy = mvf2/2 − mvi2/2

Divide by m and multiply by 2:

2gy = vf2 − vi2

Rearrange and take the square root:

vf = √(2gy + vi2)

This is the same equation for the velocity of an object projected downward.

When an object is simply dropped, vi = 0 and the equation becomes:

vf = √(2gy)

## Work when object projected upward

The work by gravity against inertia when an object is projected upward only occurs when the object starts falling downward. On the way up, you are doing work against gravity.

(See Work Against Gravity and Inertia by an External Force for more information.)

The work done depends on whether you measure the work from the peak or maximum displacement or from the starting point where the object was released.

### Work measured from maximum displacement

When an object is projected upward at some initial velocity, it will reach a maximum displacement before falling downward and doing work against inertia.

Note: The initial velocity is the velocity at which the object is released after being accelerated from zero velocity. Initial velocity does not occur instantaneously.

The equation for the maximum displacement is:

ym = −vi2/2g

where ym is the maximum displacement from the starting point in meters (m) or feet (ft)

Note: According to our convention for directions, displacements above the starting point are negative and thus ym < 0. Also, upward velocities are negative and thus vi < 0.

Work measured from the maximum displacement is simply work done by a falling object:

W = mgy

### Work measured from starting point

When you project an object upward, you are doing work against gravity as a result of your initial velocity. Once the object starts falling downward, you can begin to measure the work gravity does to overcome inertia.

#### Above starting point

While the object is moving downward above the starting point, the work done by gravity with respect to the starting point is negative:

W = −mgy

Since the direction of y is in the opposite direction of gravity, it is a negative number, according to our convention for directions.

You can also see that ΔPE is negative, since hf > hi.

Likewise, ΔKE is negative, since vf2 < vi2.

#### Below starting point

As the object travels below the starting point, the work done by gravity is the same as if the object had been projected downward at a positive value of the initial velocity.

## Summary

The product of the force of gravity and the displacement moved is the work done by gravity against inertia. Work is also the change in the potential energy or the change in the kinetic energy of the object.

Equating the change in energies allows you to calculate the final velocity of the object after it moves a given displacement.

When the object is projected upward, the work above the starting point is negative and equals zero when the object returns to the starting point. Afterwards, it follows the standard equations.

Be conscientious

## Resources and references

Author's Credentials

### Websites

Work by gravity by Sunil Kumar Singh - Connexions

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

Gravity Resources

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