Dimensions of Work

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Introduction to Work Done

Work and energy are the terms that are used in day-to-day life. Work is used to describe any kind of activity that involves physical and mental effort.

In physics, work is not defined in this way.

If we push or pull a load, or ft an object above the floor, we are doing work. However, a man carrying a heavy load and standing still at a place is not doing any work according to physics.

Energy is also a term that is often used. Energy is associated with work. The capacity of an object or a person to perform work is called energy.

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Energy is the capacity to do work.

Work was Done Definition & Formula

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

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

  3. 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.

  4. 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 transfer of the motion of an object occurs due to some applied force. Work is represented by the product of force and displacement.

  1. 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. 

  2. 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.

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 transfers energy from one place to another or from one form to another

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

W=F . ∆x 

This can also be written as:

W=F. lx 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.

1J=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 = Δv/t = Δvt

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 LT-2].

FAQ (Frequently Asked Questions)

Q1. How Do You Explain the Unit of Work?

Ans: 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. 

Q2. A force of 3 N Acts through a Distance 3 m in the Direction of the Force. Calculate the Work Done by the Force?

Ans: Here Force acting on the object and displacement if the object happens in the same direction.  

So, using the formula for work done,

W = F * d

= 3 × 3

= 9 J

Q3. A girl is Dragged Along by a Man, by Making an Angle 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.

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Ans: Here,

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

W = F * d cosθ 

= 40 × 50 × 1 = 2000 JW

Q4. What are Some Non-SI Units of Work?

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