Kinematics of a triangular three-link system — three rigid rods hinged together to form a triangle undergo general motion (rotation + translation). How many instantaneous centres of rotation exist for the triangle?

Difficulty: Easy

Correct Answer: 3

Explanation:


Introduction / Context:
Instantaneous centres (ICs) are points about which a link has a pure rotation at a given instant. For mechanisms, the number and location of ICs aid in velocity analysis without calculus. The Kennedy–Aronhold theorem governs the count and collinearity of ICs among multiple links in plane motion.


Given Data / Assumptions:

  • The system consists of three rigid links forming a triangle with pin joints.
  • Plane motion is considered (combination of translation and rotation).
  • No slipping at pins; ideal hinges.


Concept / Approach:

For a mechanism with n links in plane motion, the number of instantaneous centres is N = n*(n − 1)/2. With three links, N = 32/2 = 3. Kennedy’s theorem further states that the three ICs for any three bodies in plane motion are collinear, aiding construction and checks in velocity diagrams.


Step-by-Step Solution:

Identify n = 3 links: two moving links plus the ground (if treated as a link) — still total three for a closed triangular set.Compute N = n(n − 1)/2 = 3.Enumerate the ICs: one for each pair of links.Use collinearity (Kennedy) when constructing the ICs geometrically.


Verification / Alternative check (if short method exists):

Alternative velocity analysis using relative velocities will reveal three pairwise relative rotations consistent with three ICs.


Why Other Options Are Wrong:

Values 1, 2 undercount; 4, 5 overcount for n = 3 in planar motion.


Common Pitfalls (misconceptions, mistakes):

Forgetting to include the frame as a link in open-chain analyses; assuming one IC suffices for composite rigid-body motion without considering relative motions between links.


Final Answer:

3

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