Talha's Physics Academy
Work and Energy - Fundamentals of Work
Q.1 Define Work, its formula and unit. Discuss special cases of Work.
Physical Definition
"Work is said to be done if a force causes a displacement in a body in the direction of force."
Alternative: "The work done by a constant force is defined as the product of the component of the force and the displacement in the direction of displacement."
Alternative: "The work done by a constant force is defined as the product of the component of the force and the displacement in the direction of displacement."
Mathematical Definition
"Work is the scalar product or dot product of the force vector and the displacement vector."
$W = \vec{F} \cdot \vec{S} = F S \cos\theta$ ---(i)
Where:
- $F$ = Magnitude of Force
- $S$ = Magnitude of Displacement
- $\theta$ = Angle between force vector ($\vec{F}$) and displacement vector ($\vec{S}$)
Equation (i) can also be written as:
$W = F (S \cos\theta)$
Where $S \cos\theta$ is the component of displacement in the direction of force.
Dimension and Nature
- Nature: Work is a scalar quantity.
- Dimension: $[M L^2 T^{-2}]$
Units of Work
- In S.I. system: Joule ($\text{J}$) or $\text{N}\cdot\text{m}$
- In C.G.S. system: Erg or $\text{dyne}\cdot\text{cm}$
- In F.P.S. system: $\text{ft}\cdot\text{lb}$ (foot-pound)
Special Cases of Work
(i) Positive Work
If force and displacement are in the same direction ($\theta = 0^\circ$ or $\theta < 90^\circ$), work done is positive.
$W = F S \cos 0^\circ = (F)(S)(1) = F S$
(ii) Zero Work
If force and displacement are perpendicular to each other ($\theta = 90^\circ$), work done is zero.
$W = F S \cos 90^\circ = (F)(S)(0) = 0$
(iii) Negative Work
If force and displacement are in opposite directions ($\theta = 180^\circ$), work done is negative.
$W = F S \cos 180^\circ = (F)(S)(-1) = -F S$
Q.2 Describe the method to determine Work from Force-Displacement Graphs.
Work Done from Force-Displacement Graph
To calculate the total work done by a variable or constant force from a force-displacement graph, you need to find the area under the graph. The total area under the curve represents the total work done.
- (i) When the graph is a straight line (Constant Force):
The work can be calculated using the simple formula:$\text{Work} = \text{Force} \times \text{Displacement}$where the force remains constant along the line.
Divide the area under the curve into standard geometric shapes such as rectangles, squares, and triangles.
For triangular regions, the area is calculated using:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
Figure: When the graph is a straight line (Constant Force).
Sum up the areas of all individual geometric shapes under the graph to find the total work done:
$\text{Total Work} = \text{Area}_{\text{I}} + \text{Area}_{\text{II}} + \text{Area}_{\text{III}}$
Figure: When the graph is not a straight line (Variable Force).
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