# Mechanical Work

by Ron Kurtus (revised 7 October 2013)

When you apply a force on an object that moves the object,

When a force acts upon an object to cause a displacement of the object, it is said that **work** was done upon the object.

* Mechanical work* (as opposed to thermodynamic work) is the product of a force applied on an object and its displacement while that force is applied. It is also a measurement of the change in energy caused by the force.

Applying a force to an object accelerates the object, overcoming the resistance from intertia.

Displacement is the straight-line distance in the direction of motion. Both force and displacements are vectors.

If you are accelerating an object, the resistance to motion is the inertia of the object. Thus, you are doing work is against inertia. Since objects tend to continue moving after a force is applied, the distance is only measured while that force is being applied.

Typically, you consider doing work against a continuous resistive force, such as friction or gravity. Unless the object is already moving, the force applied must be greater than the resistive force.

Questions you may have include:

- What is the relationship of work to force?
- What is work against inertia?
- What is work against a resistive force?

This lesson will answer those questions. Useful tool: Units Conversion

## Work is force times displacement

A definition of mechanical work is that it equals the force to *overcome a resistance* times the distance or displacement traveled in the direction of the force and while that force is being applied. The equation is:

W = Fd

where:

**W**is the work in joules (J or kg-m²/s²) or foot-pounds**F**is the force required to move the object in newtons (N or kg-m/s²) or pounds (lbs)**d**is the displacement of the object in meters (m) or feet (ft)**Fd**is**F**times**d**

Note: In this contextWindicates work. SometimesWis also used for weight. You should always make sure you read the definition below the equation to make sure you understand what the letters stand for.

Note:dis used here for distance or displacement, instead ofxory, because the direction doesn't matter. Some books usesfor distance.

### Force at an angle

If the force moving the object is at an angle with the direction of the resistance, the equation becomes:

W = Fd(cos θ)

where **θ** is the angle between the direction of the force and the direction of motion.

One application of this is when moving an object up a ramp against gravity. But note that the resistance of inertia is always in line with the force.

### When resistance too great

If a force is applied on an object and there is no movement because the resistance is too great, then there is no work. If you push on a heavy object but are unable to move it, you are making an effort but you are not doing any work, according to the scientific definition of work.

## Work against inertia

When you apply a force on a stationary but freely moving object, you are working against its inertia or tendency to remain stationary. This also applies to changing the velocity or direction of an object.

Note: Newton's first law of motion is often called theLaw of Inertia. It states that:

Matter remains in its state of motion and direction unless acted upon by a force.(See

Motion and the Law of Inertiafor more information.)

### Force required

The force required to move an object against inertia is:

F = ma

where

**m**is the mass of the object**a**is the resulting acceleration

Thus, the work against inertia is:

W = mad(Don't get "mad" about this!)

The work done on a freely moving object only occurs over the distance* while you are applying the force*. Once you stop applying the force, the object moves freely and no more work is being done.

### Example of throwing a ball

For example, if you throw a ball, the work done consists of the distance you accelerated the ball until you let it go. Once you have thrown the ball, it will continue at a constant velocity (minus the effect of air resistance) and no further work is done.

### Example of carrying a heavy box

If you are holding a heavy box and then carry it across the room, the work you are doing against inertia is the force you apply to move the box (**F = ma**) times the distance you carry it.

Notethat some textbooks say that this isnotwork, because the force of gravity is perpendicular to your motion. They neglect the effect of inertia on preventing the acceleration of the box.

Moving the box across the room is **work against the inertia** of the box, while lifting the box up is **work against the resistive force of gravity**.

## Work against a resistive force

A resistive force is a force that causes a moving object to slow down or tends to prevent a stationary object to move. The resistive force acts in a direction opposite to the one that you want to move the object.

Just as going against inertia, the distance is only measured while the force is applied, since it is possible for an object to continue moving a short distance after the force is released, even though it is moving against a resistive force.

### Example: Work against gravity

When you lift a heavy object, you are doing work against the force of gravity. However, the amount of work depends on the initial motion of the object.

#### Object already moving upward

If the object is already moving upward at some velocity, the force required to lift it further at the same velocity is:

F = mg

where

**F**is the force required to lift the object and is also its weight**m**is the mass of the object**g**is the acceleration due to the force of gravity (9.8 m/s² or 32 ft/s²)

The work required to lift the object a certain distance at the same initial velocity is:

W = mgd

#### Object initially stationary

The usual situation in lifting an object is that it is initially stationary. In such a case, you must not only overcome the force of gravity, but you must also overcome inertia in accelerating the object.

The work required is a sum of the work against gravity and against inertia:

W = mgd + mad

W = md(g + a)

For example, if you lift a stationary object with a mass of **0.5 kg** a distance of **1m** with an acceleration of **1.2 m/s²**, the work done is:

W = (0.5 kg)(1 m)(9.8 m/s² + 1.2 m/s²)

W = 5.5 kg-m²/s²

Note: If you accelerate the object in lifting it and then decelerate back to zero velocity, the work against inertia cancels out and is zero.

### Work against other resistances

A similar logic follows in doing work against other resistive forces, such as friction, air resistance, and spring resistance.

However, they are not as straightforward as in the gravity case. For example with friction, you may have to consider both static friction and kinetic friction in calculating the work done against friction.

## Summary

Work is the result of a force moving an object a distance, measured while that force is being applied. The equation for work is **W = Fd**. Work can be to overcome inertia, as well as to work against a resistive force. Gravity can do work against inertia and you may do work against the force of gravity.

Work hard to achieve your goals

## Resources and references

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**Top-rated books on Physics of Work**

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## Work is a Result of Force