Question

Given below are two statements:
Statement (I): The piston attains its maximum velocity at top dead centre.
Statement (II): The piston attains its maximum acceleration at top and bottom dead centre.

In light of the above statements, choose the most appropriate answer from the options given below.

• A. Both Statement (I) and Statement (II) are true.
• B. Both Statement (I) and Statement (II) are false.
• C. Statement (I) is true but Statement (II) is false.
• D. Statement (I) is false but Statement (II) is true.

Hint:

At extreme travel limits (TDC/BDC): Reciprocating velocity is zero, and acceleration is at its maximum peak because the piston is reversing direction.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Piston motion in a slider-crank mechanism is a form of reciprocating motion characterized by varying velocity and acceleration throughout the stroke.

Step 2: Key Formula or Approach:
For a crank radius \(r\), connecting rod length \(l\), and crank angle \(\theta\) (measured from TDC):
The piston velocity \(v\) is:
\[ v \approx \omega r \left( \sin\theta + \frac{\sin 2\theta}{2n} \right) \] The piston acceleration \(a\) is:
\[ a \approx \omega^2 r \left( \cos\theta + \frac{\cos 2\theta}{n} \right) \] where \(n = l/r\).

Step 3: Detailed Explanation:
1.

Statement (I): At the Top Dead Center (TDC, \(\theta = 0^\circ\)) and Bottom Dead Center (BDC, \(\theta = 180^\circ\)), the piston must momentarily stop to reverse its direction of travel.
Thus, the velocity of the piston at TDC is exactly zero. Maximum velocity is achieved near the middle of the stroke (\(\theta \approx 90^\circ\)). Therefore, Statement (I) is false.
2.

Statement (II): Evaluating the acceleration equation:
- At TDC (\(\theta = 0^\circ\)):
\[ a_{\text{TDC}} = \omega^2 r \left(1 + \frac{1}{n}\right) \] - At BDC (\(\theta = 180^\circ\)):
\[ a_{\text{BDC}} = -\omega^2 r \left(1 - \frac{1}{n}\right) \] These represent the maximum positive and negative acceleration peaks in the cycle. Therefore, Statement (II) is true.

Step 4: Final Answer:
The correct option is 4, which corresponds to Statement (I) is false but Statement (II) is true.