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# Dimensions of Work     ## An Introduction to Work

Last updated date: 27th Mar 2023
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Work and energy are the two interrelated casual terms that are used in the day-to-day life of every living thing to simply survive in this world. Work is used to describe any kind of activity that involves physical and mental effort and to perform any work you need energy.

But in physics, work is not defined in this way. If we push or pull a load, or fit an object above the floor, it is considered work is performed there. However, when a man is carrying a heavy load but standing still and stationary at a place and not doing any work or when there is no movement according to physics is not defined as work. Energy is also a term that is often used commonly and is closely associated with work. The capacity of an object or a person to perform any particular work is called energy.

### Work Done Definition & Formula

• Work is said to be done when the force applied to an object produces motion in the object.

• Work can be defined in terms of force applied to an object and the displacement of the object.

• Consider a book placed on a frictionless horizontal table. The book is acted upon by a constant force F. The action of this force is to move the book through a distance d in a straight line in the direction of the applied force.

• Work done through the application of force is equal to the product of the magnitude of the applied force and displacement of the object. Mathematically work done by a force is given by the formula:

W = F * d

Where F = |F|

### What is the Dimension of Work?

Physical quantities are measured in terms of some units; some of the units are expressed in terms of length and time. In this manner, the area is represented as the square of the length, the volume is represented as a cube of length, velocity is represented as length over distance, etc.

Work is a basic unit, and it is not described in terms of length and time. The dimension of work is the same as that of the dimension of energy.

Electric charge, mass, temperature, is also fundamental quantities that cannot be expressed in terms of other units.

The units that we use are related by the physical formulae, and for the sake of convenience, it is decided which units should be used to calculate dimensions.

In physics, work is a process in which the transfer of the motion of an object occurs due to some applied force. Work is represented by the product of force and displacement.

• Work done is said to be positive if the force applied on an object displaces the object in the direction of force, and hence the force does positive work.

• Work done is taken as negative when the force applied on an object displaces the object in a direction opposite to the applied force, and hence the force does negative work.

### Calculation of Dimension of Work

In order to calculate the dimension of any physical quantity, it is necessary to consider the definition and formula of the quantity with the known dimensions.

Work is a scalar quantity. It is a scalar product of force and displacement. The displacement of the object may or may not be due to the applied force.

For example, work is applied to throw a ball upwards, but when the ball returns to the ground, a gravitational force acts on it rather than the actual applied force. A force that is equal to the ball’s weight acts on it.

A constant force F is applied to an object causing a displacement s in the object, and if the angle between force and displacement is θ, then the work done by the force is given by,

W = Fs cosθ.

Work was done to an object either displace it or transfer energy from one place to another or from one form to another

The SI unit of work is Joule, also depicted as J.

$W = \overrightarrow{F} . \overrightarrow{\Delta x}$

This can also be written as:

W=F. l $\overrightarrow{\Delta x}$ l.cos⁡(θ),

Where,

θ = angle between the force and the displacement vector.

The cosine of the angle is dimensionless. The dimension of displacement is that of length, L, or meters.

The force is measured in the units of Newtons, which is a unit composed of other fundamental units:

$1N = kg.\frac{m}{s^2}$

Therefore, the dimension of force is the dimension of work.

In the metric system, work is measured in the SI unit Joule.

1 J = N.m = Kg $\frac{m}{s^2}$

The Dimension of Work Done

Work = Force × Displacement

i.e., W = F × s

Where,

s - Displacement

F - Force applied

W - Work is done by the force

So, the dimension of displacement is = L1

Let’s derive the dimension of Force:

F = m.a

Where,

m = mass of the object

a = Acceleration produced in the object

The dimension of Mass = [M1]

Now acceleration

$a = \frac{\Delta v}{t} = \Delta v t^{-1}$

The dimension of Velocity is = [M0L1T−1]

The dimension of Time = [M0T1]

So, the dimension of acceleration is given by

= [M0L1T−1] ÷ [M0T1]

= [M0L1T−2]

So, the dimension of force is given by

Dimension of Force =  [ M1] × [M0L1T−2] = [M1L1T−2]

Since we have found out the dimensions of both force and displacement, we can easily calculate the dimension of work.

The dimension of Work = dimension of force × dimension of displacement

= [M1L1T−2] × [L1]

= [M1L2T−2]

= Unit of work done is Joule.

Therefore, the dimension formula of work is represented as [M1 L2 T-2].

### Conclusion

Hence the above article provides information about work done and its definition and dimension. The dimension of the work done is derived using dimensions of force and displacement.

## FAQs on Dimensions of Work

1. How Do You Explain the Unit of Work?

SI unit of work is joule (J). One joule is defined as the work done by a force of one Newton which causes a displacement of one meter in the object.

Newton-Meter is dimensionally equivalent to the dimension of work. It is also sometimes used as a unit of work.

2. A force of 3 N acts through a distance of 3 m in the direction of the force. Calculate the work done by the force?

Here force acts on the object and displacement if the object happens in the same direction.

So, using the formula for work done,

W = F $\times$ d

= 3 $\times$  3

= 9 J

3. A girl is dragged along by a man, by making an angle of 45° with the horizontal. The force applied by the rope is 40 N and the box is dragged by 50 m. Calculate the work done by the force.

Here,

Force and displacement are at an angle of 45°. So, work done by the force

W = F $\times$ d cosθ

= 40 $\times$ 50 $\times$ 1 = 2000 JW

4. What are Some Non-SI Units of Work?

The non-SI units of work are Newton-Meter, erg, foot-pound kilowatt-hour horsepower, etc.

5. How is the Dimension of work calculated?

To calculate the dimension of any physical quantity involved, it is necessary to consider the definition and formula of the quantity with the known available dimensions. Work is a scalar quantity and a scalar product of force and displacement. The displacement or any movement in the object may or may not necessarily be due to the applied force. For example, work is applied to throw a ball upwards in the air, but when the same ball returns to the ground due to the gravitational force acting on it rather than the actual force applied externally. Therefore work is a force that is equal to the ball’s weight acting on it.

6. What are the units of work involved?

Whenever any form of a new quantity or new terms is introduced in physics, the standard metric units associated with that particular quantity are discussed in detail and explained very clearly. As in the case of work (and also energy), the standard metric unit is the Joule which is abbreviated as J. Generally it is calculated as one Joule is equivalent to one Newton of force causing a displacement of one meter. In other words,

The Joule is the unit of work.

1 Joule = 1 Newton $\times$ 1 meter

1 J = 1 N $\times$ m

7. How does the term work differ in physics?

When any force acts upon an object to cause a displacement of or any movement in the object, it is said that work was done upon the object. There are three main ingredients to work: force, displacement, and cause. For any particular force to qualify as having performed work on an object, there must be a proper movement or displacement and that particular force must be caused by the displacement. There are several good examples of work that can be observed in our everyday life - a horse pulling a plough through the field, a father pushing a grocery cart down the aisle of a grocery store, etc. In each case described here, there is an external force exerted upon an object to cause that object to be displaced or moved from its original condition.

8. What are the 3 main variables of work available?

The equation for work lists three variables - each variable is associated with these terms force, displacement, and cause. The angle theta in the equation is associated with the exact amount of force that causes a specific amount of movement or displacement caused to the object. When a force is generally exerted on any particular object at an angle to the horizontal, only that part of the force contributes to or causes a horizontal displacement to that specific object.

9. How is the term Work involved in our daily lives?

In physics, work is generally the consideration and involvement of the total amount of energy transferred to or from an object via the application of force along with a movement or displacement. In its simplest form, it is often represented as the product which involves force and displacement. Work is done whenever a force moves something from its original place. Everyday examples of work in our daily life include walking upstairs, lifting heavy objects, pulling a sledge, and pushing a shopping trolley. Whenever work is done, energy is transferred from one place to another effectively to move them.