To find the length of the rocker link $ZB$ for this four-bar design, given that the stroke length $BB'$ is $49 \, \text{cm}$ and the angle $\theta$ is $50^\circ$, we can use the geometry of the setup.

Solution:

1. The stroke length $BB'$ is the distance that point $B$ moves when the rocker swings through the angle $\theta$.

2. Since $B$ traces an arc of a circle centered at $Z$, we can model $BB'$ as the chord length in a circle with radius $ZB$.

3. The formula for the chord length $BB'$ in terms of the radius $r = ZB$ and angle $\theta$ (in radians) is: $BB' = 2 \cdot ZB \cdot \sin\left(\frac{\theta}{2}\right)$

4. Converting $\theta = 50^\circ$ to radians: $\theta = 50^\circ \times \frac{\pi}{180} = \frac{5\pi}{18} \approx 0.8727 \, \text{radians}$

5. Now, solve for $ZB$ using the given chord length $BB' = 49 \, \text{cm}$: $49 = 2 \cdot ZB \cdot \sin\left(\frac{0.8727}{2}\right)$

6. Calculating $\sin\left(\frac{0.8727}{2}\right)$: $\sin\left(\frac{0.8727}{2}\right) \approx \sin(0.43635) \approx 0.4226$

7. Substitute and solve for $ZB$: $49 = 2 \cdot ZB \cdot 0.4226$ $ZB = \frac{49}{2 \cdot 0.4226} \approx \frac{49}{0.8452} \approx 57.97 \, \text{cm}$

Answer:

The length of the rocker link $ZB$ is approximately $57.97 \, \text{cm}$, rounded to two decimal places.

Would you like a further breakdown of any steps?

Here are some related questions for further practice:

1. How would the result change if $\theta$ were larger or smaller?
2. Can you calculate the length of $ZB$ if the stroke length $BB'$ is 60 cm, with the same angle $\theta$?
3. How would you find $BB'$ if $ZB$ and $\theta$ were given instead?
4. What is the effect on the four-bar mechanism’s movement if $ZB$ is increased?
5. If $\theta$ was given in radians initially, how would you set up the problem?

Tip: Always ensure angles are in radians when using trigonometric functions in physics and engineering calculations.