For an inline slider-crank mechanism, the length of the crank and connecting rod are 3 m and 4 m, respectively. At the instant when the connecting rod is perpendicular to the crank, if the velocity of the slider is 1 m/s, the magnitude of angular velocity (upto 3 decimal points accuracy) of the crank is _________radian/s.Correct answer is between '0.26,0.27'. Can you explain this answer?

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Accepted Answer

Answer

Given Data:

Length of the crank (l) = 3 m

Length of the connecting rod (r) = 4 m

Velocity of the slider (v) = 1 m/s

Understanding the Problem:

In an inline slider-crank mechanism, the crank and connecting rod are connected at one end, and the slider is attached to the other end of the connecting rod. The crank rotates, and the slider moves back and forth in a straight line.

Approach:

To find the angular velocity of the crank, we need to determine the angle between the connecting rod and the crank when the connecting rod is perpendicular to the crank. We can then use the velocity relationship between the slider and crank to calculate the angular velocity.

Calculation:

Let's consider the crank as the base and the connecting rod as the moving part.

When the connecting rod is perpendicular to the crank, it forms a right-angled triangle with the crank. Let's label the angle between the crank and the connecting rod as θ.

Using the Pythagorean theorem, we can find the distance (d) between the base and the slider:

d² = (l - r*cosθ)² + (r*sinθ)²

Since the velocity of the slider is given as 1 m/s, we can relate the angular velocity (ω) of the crank and the linear velocity (v) of the slider:

v = ω * d

Substituting the value of d from the previous equation:

1 = ω * sqrt((l - r*cosθ)² + (r*sinθ)²)

Squaring both sides of the equation:

1 = ω² * ((l - r*cosθ)² + (r*sinθ)²)

Expanding and simplifying the equation:

1 = ω² * (l² + r² - 2*l*r*cosθ)

Since we know that the connecting rod is perpendicular to the crank, cosθ = 0:

1 = ω² * (l² + r²)

Solving for ω:

ω = sqrt(1 / (l² + r²))

Substituting the given values of l and r:

ω = sqrt(1 / (3² + 4²))

ω = sqrt(1 / (9 + 16))

ω = sqrt(1 / 25)

ω = 1/5 = 0.2 rad/s

Conclusion:

The magnitude of the angular velocity of the crank is 0.2 rad/s, which is not within the given range of 0.26 to 0.27 rad/s. Therefore, there might be an error in the given data or calculations. Please review the problem statement and double-check the calculations.

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