To determine the crank length ($ZB$) in a four-bar mechanism where the rocker $ZB$ has a length of 28 cm and moves between positions $B$ and $B'$, forming an angle $\theta = 56^\circ$, we can use trigonometry. Since the rocker moves in an arc, we can use the formula for the horizontal distance covered based on this angular motion.

Solution:

1. The given information states:

   • Rocker length ($ZB$) = 28 cm
   • Angle $\theta = 56^\circ$

2. To find the crank length ($ZB$), which corresponds to the horizontal distance between $B$ and $B'$ when the rocker is at $B$ and $B'$, we can calculate the length of the arc movement at angle $\theta$.

3. The effective crank length can be calculated using the following formula:

   $ZB' = ZB \cdot \sin\left(\frac{\theta}{2}\right)$

4. Plugging in the values: $ZB' = 28 \cdot \sin\left(\frac{56^\circ}{2}\right)$

Calculation:

First, we calculate $\frac{56^\circ}{2} = 28^\circ$.

Then, we compute $\sin(28^\circ)$: $ZB' = 28 \cdot \sin(28^\circ) \approx 28 \cdot 0.4695 = 13.15 \text{ cm}$

Answer:

The crank length $ZB'$ is approximately 13.15 cm (rounded to two decimal places).

Would you like further details or have any questions?

Here are five related questions to deepen understanding:

1. What is a four-bar mechanism, and what applications does it have in engineering?
2. How would changing the angle $\theta$ affect the calculated crank length?
3. Why is the sine function used in this calculation instead of cosine or tangent?
4. What would happen to the mechanism's range of motion if the rocker length were doubled?
5. Can this type of calculation be applied to other types of mechanical linkages?

Tip: In trigonometric calculations involving angles, always ensure your calculator is set to the correct mode (degrees or radians) as per the problem’s requirements.