Question

In a given mechanism, the number of links are five. Then the number of instantaneous centres of the mechanism is

• A. $10$
• B. $6$
• C. $5$
• D. $15$

Hint:

This formula counts how many unique pairs you can make from $n$ items. For common mechanisms, the number of I-centers increases rapidly as links are added: - 4 links $\rightarrow$ 6 I-centers - 5 links $\rightarrow$ 10 I-centers - 6 links $\rightarrow$ 15 I-centers

Answer 1

The Correct Option is A

Concept: In the kinematic analysis of mechanisms, an instantaneous center (I-center) is a virtual point common to two bodies that has the same instantaneous velocity in both magnitude and direction. For a kinematic mechanism composed of a fixed number of rigid links, the total number of instantaneous centers $N$ corresponds to the number of unique pairs that can be formed from those links. Mathematically, this is found using the combinations formula $N = \binom{n}{2}$: \[ N = \frac{n(n - 1)}{2} \] where $n$ represents the total number of links inside the mechanism.

Step 1: Evaluating the formula for the given number of links.
The problem states that the mechanism contains five links: \[ n = 5 \] Substitute $n = 5$ into our combinations equation: \[ N = \frac{5 \times (5 - 1)}{2} \]

Step 2: Simplifying the arithmetic computation.
\[ N = \frac{5 \times 4}{2} = \frac{20}{2} = 10 \] Thus, a 5-link mechanism contains exactly $10$ unique instantaneous centers of rotation. This matches Option (A).