Unit 20
Mechanics
Class 12 Mathematics
Parallel Forces
Class 12 Mathematics – Parallel Forces Notes PDF
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This mechanics chapter focuses on resultants of like and unlike parallel forces and the moment concepts used to locate their lines of action. Closely related statics ideas—Varignon’s theorem and couples—are included for exam-ready understanding.
1. Parallel Forces
Parallel forces are coplanar forces whose lines of action are parallel. They may act in the same direction (like) or opposite directions (unlike).
2. Resultant of Like Parallel Forces
For two like parallel forces P,Q, the resultant magnitude is
The resultant acts between the two forces. If their separation is d and R is at distance x from P, moment balance gives
Thus the resultant divides the distance inversely in the ratio of the forces.
3. Resultant of Unlike Parallel Forces
If P>Q and they act oppositely,
The resultant acts outside the two forces on the side of the larger force. Its position is found by equating moments.
For upward 10 N and downward 4 N separated by 3 m, resultant is 6 N upward. If its distance from the 10 N force is x outside the interval:
6x=4×3 ⇒ x=2 m.
4. Moment of a Force
The moment of a force about a point equals force × perpendicular distance from the point to its line of action.
Clockwise and anticlockwise moments are assigned opposite signs under a chosen convention.
5. Varignon’s Theorem
The moment of the resultant about any point equals the algebraic sum of the moments of the component forces about that point.
M_O(R)=ΣM_O(Fᵢ).This theorem is the main tool for locating a resultant line of action.
6. Couple and Moment of a Couple
A couple consists of two equal, opposite, parallel forces whose lines of action are separated.
where d is the perpendicular distance between the forces. A pure couple has zero resultant force but a non-zero rotational effect.
7. Worked Examples
For 6 N and 4 N upward separated by 5 m: R=10 N. Distance from 6 N force: x=4×5/10=2 m.
A 20 N force acts 0.3 m from a pivot. Moment = 20×0.3=6 N·m.
8. Common Mistakes
For unlike forces, subtract magnitudes and place the resultant outside.
Always use perpendicular distance in a moment.
Choose and keep a consistent clockwise/anticlockwise sign convention.
A couple has zero resultant force but non-zero moment.
9. Important Exam Questions
- Find magnitude and position of resultant of two like parallel forces.
- Solve an unlike parallel-force problem.
- Define and calculate moment of a force.
- State/apply Varignon’s theorem.
- Define a couple and compute its moment.
Discussion
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