To calculate the mass $m$, given angles and force, we need to interpret the situation. It appears this problem involves forces and angles, likely in a system like an inclined plane, a pulley system, or some other force balance setup. Let's assume it is a situation involving two forces acting at angles, possibly with tension in a rope, which is a common setup.

Assumptions:

1. $\theta_L$ and $\theta_R$ are the angles at which the forces are acting.
2. $F_R$ is the force acting on the right side of the system.
3. We need to balance the forces and use trigonometry to find mass $m$, where the force on the left is due to the weight $mg$, and $g = 9.8 \, \text{m/s}^2$.

Step-by-Step Solution:

The forces acting can be resolved into horizontal and vertical components. The vertical force on the left side is the weight $mg$, while the horizontal force is balanced by the tension forces at angles $\theta_L$ and $\theta_R$.

For equilibrium in the vertical direction: $F_L \sin(\theta_L) = F_R \sin(\theta_R)$ And since $F_L = mg$, we substitute this: $mg \sin(\theta_L) = F_R \sin(\theta_R)$ Now solve for mass $m$: $m = \frac{F_R \sin(\theta_R)}{g \sin(\theta_L)}$

Calculation:

Using the given values:

• $\theta_L = 31.40^\circ$
• $\theta_R = 17.20^\circ$
• $F_R = 86.4 \, \text{N}$
• $g = 9.8 \, \text{m/s}^2$

We can substitute these values and compute $m$.

Let me calculate that for you:The mass $m$ is approximately $5.00 \, \text{kg}$.

Would you like further details or clarifications? Here are five related questions to explore further:

1. How do trigonometric functions apply to force balance problems?
2. What is the significance of resolving forces into horizontal and vertical components?
3. How would changing the angles affect the mass in this system?
4. What if friction were introduced—how would that affect the calculation of mass?
5. Can you derive the formula used to solve this problem?

Tip: Always ensure angles are in the correct unit (radians vs degrees) when using trigonometric functions in calculations.