https://file.aiquickdraw.com/mathbot/file/1730439276249s290by6r.jpg

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.

For a four-bar design, calculate the rocker link length in cm so that the stroke length BB' is 49 cm and the angle θ is 50 degrees. Write the value up to 2 places of decimal.

Learn how to calculate the rocker link length in a four-bar mechanism design with a stroke length of 49 cm and angle of 50 degrees. This problem involves trigonometric formulas for chord length and angle conversion, making it ideal for undergraduate engineering students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation