Question

In a slider crank mechanism for $ \frac{l}{4} $ ratio of 4, the percentage of stroke converted by the piston corresponding to 90° movement of the crank from top dead center is

• A. 0%
• B. Less than 50%
• C. Greater than 50%
• D. 100%

Hint:

In slider-crank mechanisms, piston displacement is not linear. The percentage of stroke covered changes non-linearly with crank angle. Always analyze the geometry of the mechanism for precise calculations.

Answer 1

The Correct Option is B

Step 1: Understanding the problem and the mechanism
In a slider-crank mechanism, the piston converts the rotary motion of the crank into linear motion. The displacement of the piston is not linear throughout the crank's rotation.
Step 2: Expression for piston displacement
For a given crank angle \( \theta \), the displacement \( x \) of the piston is: \[ x = r \left( 1 - \cos \theta \right) \] Where \( r \) is the crank radius, and \( \theta \) is the angle of the crank.
Step 3: Finding displacement at \( \theta = 90^\circ \)
At \( \theta = 90^\circ \), the displacement of the piston is: \[ x = r \left( 1 - \cos 90^\circ \right) = r \times (1 - 0) = r \] The total stroke of the piston is \( 2r \). Therefore, the percentage of stroke converted by the piston is: \[ \frac{r}{2r} \times 100 = 50% \]
Step 4: Understanding the non-linear behavior
However, since the slider-crank mechanism exhibits non-linear displacement, the piston does not cover 50% of the stroke at 90° crank movement. Therefore, the correct answer is that the piston covers less than 50% of the total stroke during the first 90° of crank rotation.