‘Work is said to be done when a net force acting on the body, displaces the body in the direction of the force.
Units of Work
The CGS unit of work is ‘erg’ ( which is derived from the Greek word ‘Energia’ meaning in work ) and the SI unit of work is joule ( J ) (in honor of the English scientist James Prescott Joule).
1 erg = 1 dyne × 1 cm
Hence, one erg is defined as the work done when a net force of one dyne displaces a body through one centimeter in its direction; Similarly, 1 joule = 1 newton × 1 meter.
Hence, one joule is defined as the work done when a net force of one newton displaces it through one meter in its direction.
| CGS unit | erg |
| SI unit | joule (J) |
| Formula for work done (W) | W = Force×Distance×cos(θ) |
| Dimensional formula for work | [W]=[M]⋅[L]2⋅[T]-2 where: M represents mass, L represents length, and T represents time. |
Work done formula derivation
The work done on a body is proportional to the net force acting on the body and the displacement produced by the force on the body.
W∝F and W∝s
‘W; ‘F’ and ‘s’ are work done, applied force and displacement of the body in the direction of force, respectively.
From the above, we get
W∝Fs
where ‘k’ is a proportionality constant and the units of force and work are defined in such a way that k = 1.
Hence, W = Fs
Thus, when one unit force applied on a body produces a displacement of one unit in the direction of force, one unit of work is said to be done.
Real life example of work done
Consider a lawn roller being pulled as shown in Fig. 1.1. The direction of applied force is in the direction of displacement. Hence, the total force applied is utilized and is responsible for the displacement of the roller. Thus, we can write work done, W = Fs.
But when the direction of applied force makes an angle ‘θ’ with the direction of displacement as shown in Fig. 1.2, the total force applied is not responsible for the movement of the roller.
Only a part or a component of force which is equal to ‘Fcosθ’ is responsible for the displacement of the roller.
Hence, in this case, work done is given as,
W = (F cosθ) s or W = Fs cosθ
So, in general we can express the work done as the product of displacement of a body and the component of force responsible for the displacement of the body; The component of force responsible for the displacement will be in the direction of displacement and generally it is ‘F cos θ’ where θ is the angle between the directions of force and displacement. Hence, the general way of expressing work is W = Fs cosθ.
The three cases of work
Work done by the system is positive or negative
Case ( i ) If θ < 90°
W = Fs cosθ ⇒ work done is positive as cosθ > 0 when 0°<θ< 90 °
Case ( ii ) If θ > 90° ⇒ work done is negative as cosθ < 0 when 90°< θ < 180°
Case (iii) If θ = 90 ° ⇒ work done is equal to zero as cos 90º = 0
Example: In uniform circular motion , work done by a centripetal force is equal to zero since centripetal force and displacement of the body are perpendicular to each other.
In the case of pulling or pushing a lawn roller, work is done on the lawn roller.
When we use a pressure cooker, the steam produced in the cooker due to pressure pushes up the weight kept on the lid where work is done by steam. Work is a scalar quantity.
Work Done by the Force of Gravity
All bodies are attracted towards the centre of Earth do to force of gravity. if a body moves short distance horizontally over the surface of the earth or in a circular path around the earth, like satellite, work done by the force of gravity zero because the displacement of the body at any instant is perpendicular to the direction of force of gravity. if our body of mass ‘m’ Falls vertically down through a height ‘h’, then the work done will be equal to mgh, i.e., the change in the gravitational potential energy.
Dimensional Formula of Work
Work = (Force) × (displacement)
W = Fs
[ W ] = [ M1 L1 T-2 ] · [ L1 ] = M1 L2 T-2
Dimensional formula of work is [ M1 L2 T-2]
Frequently Asked Questions – FAQs Related to Work
The SI unit of work is the joule (symbol: J). It is named after the English physicist James Prescott Joule. One joule is defined as the amount of work done when a force of one newton is applied over a distance of one meter in the direction of the force. Mathematically, work (W) is calculated as the product of force (F) and displacement (d), multiplied by the cosine of the angle (θ) between the force and the displacement: W = F × d cos(θ) In the SI system, work and energy are measured in joules, and they are equivalent since energy is the capacity to do work. Therefore, the unit joule is also used to measure energy.
work done is maximum when the force and displacement are in the same direction, meaning the angle (θ) between them is zero degrees (θ = 0°) or when the cosine of the angle is equal to 1. In the equation for work: W = F ×d cos(θ) When θ = 0°, cos(θ) = 1. Therefore, the term cos(θ) becomes 1, and the work equation simplifies to: W = F ×d This shows that the work done is maximum when the force and displacement are parallel or collinear, and there is no angle between them. In this case, the entire force is applied in the direction of the displacement, resulting in the maximum amount of work.
To derive the relation between the newton (N) and the dyne (dyn), we need to use the conversion factor between the two units. The newton is the SI unit of force and is defined as the force required to accelerate a mass of one kilogram by one meter per second squared (N = kg * m/s^2). The dyne, on the other hand, is a unit of force in the centimeter-gram-second (CGS) system. It is defined as the force required to accelerate a mass of one gram by one centimeter per second squared (dyn = g * cm/s^2). To convert between the two units, we can use the following relation: 1 N = 10^5 dyn This conversion factor ari.ses from the definition of the newton in terms of kilograms and meters, compared to the definition of the dyne in terms of grams and centimeters. Since there are 1000 grams in a kilogram and 100 centimeters in a meter, the conversion factor is: 1 N = (1000 g) * (100 cm/s^2) = 10^5 dyn Therefore, 1 newton is equal to 10^5 dynes.
Yes, that is correct. In the SI (International System of Units), work and energy are measured using the same unit, which is the joule (J). Both work and energy involve the transfer or transformation of energy. Work is defined as the product of the force applied to an object and the displacement of that object in the direction of the force. It represents the energy transferred to or from an object by the application of a force. The SI unit of work, as mentioned, is the joule. Energy, on the other hand, is a scalar quantity that represents the ability or capacity to do work. It exists in different forms such as kinetic energy, potential energy, thermal energy, etc. The SI unit of energy is also the joule. Since work involves the transfer of energy, they share the same unit. When work is done on an object, energy is transferred to that object, and when work is done by an object, energy is transferred from that object. So, whether we are talking about the work done on an object or the energy possessed by an object, we use the unit joule to quantify them in the SI system.
No work is done on a body when the force applied to the body and the displacement of the body are perpendicular to each other. In other words, when the angle (θ) between the force vector and the displacement vector is 90 degrees (θ = 90°), the cosine of 90 degrees is zero (cos(90°) = 0), which means the work done is zero. In the equation for work: W = F * d * cos(θ) When θ = 90°, cos(θ) = 0. Therefore, the term cos(θ) becomes 0, and the work equation simplifies to: W = F * d * 0 = 0 This means that if the force applied to a body is perpendicular to the direction of its displacement, no work is done on the body. The force may be exerted on the body, but if it is not in the same direction as the displacement, there is no transfer of energy resulting in work being done. For example, if you push a box horizontally along a surface, and the force you exert is perpendicular to the displacement of the box, no work is done on the box. The force you apply may change the box’s direction, but it does not result in a transfer of energy over the displacement. Work is only done when the force and displacement have a component in the same direction.
Comments
Post a Comment