Sley Motion Analysis at Extreme Positions

The question asks about the sley velocities and accelerations at the front centre ($0^\circ$) and back centre ($180^\circ$) of a shuttle loom. These represent the endpoints of the sley's reciprocating motion.

Velocity Comparison

At the extreme ends of its travel (front centre and back centre), the sley momentarily stops before reversing direction. In simplified models of such mechanisms, like simple harmonic motion, the velocity at these extreme points is zero.

• Velocity at front centre ($v_{0^\circ}$) = 0
• Velocity at back centre ($v_{180^\circ}$) = 0

Therefore, the sley velocities are the same at both positions.

Acceleration Comparison

Acceleration is the rate of change of velocity. While the velocity is zero at both extreme points, the sley must change direction rapidly at these points.

• At the front centre ($0^\circ$), the sley stops its forward motion and begins moving backward. This requires a significant change in velocity, indicating maximum acceleration (or deceleration, depending on perspective). Let's denote this as $a_{0^\circ}$.
• At the back centre ($180^\circ$), the sley stops its backward motion and begins moving forward. This also requires a significant change in velocity, indicating maximum acceleration (or deceleration). Let's denote this as $a_{180^\circ}$.

Using a simple harmonic motion model $x(t) = A \cos(\omega t)$, velocity is $v(t) = -A \omega \sin(\omega t)$ and acceleration is $a(t) = -A \omega^2 \cos(\omega t)$.

• At $0^\circ$ ($\omega t = 0$), $v=0$ and $a = -A \omega^2$.
• At $180^\circ$ ($\omega t = \pi$), $v=0$ and $a = -A \omega^2 \cos(\pi) = A \omega^2$.

The velocities are the same (zero), but the accelerations have the same magnitude ($A \omega^2$) but opposite directions ($-A \omega^2$ and $A \omega^2$). Thus, the accelerations are different.

Conclusion

Based on the analysis of motion at the extreme points:

• Sley velocities are the same (zero).
• Sley accelerations are different (equal magnitude, opposite direction).

This corresponds to Option 1.