Let's break down the two parts of this problem involving the crane arm's torque and load.

Part 1:

We are given the following information for part 1:

• Length of the crane arm: $L = 19.0 \, \text{m}$
• The angle between the crane arm and the horizontal: $\theta = 13.2^\circ$
• Maximum load the crane can handle: $F = 706 \, \text{N}$

The torque ($\tau$) is calculated using the formula: $\tau = F \cdot L \cdot \sin(\theta)$ where:

• $\tau$ is the torque in $\text{N} \cdot \text{m}$,
• $F$ is the force (maximum load) in newtons,
• $L$ is the length of the crane arm in meters,
• $\theta$ is the angle in degrees.

Substituting the given values: $\tau = 706 \, \text{N} \times 19.0 \, \text{m} \times \sin(13.2^\circ)$

We will calculate this next.

Part 2:

For part 2, we are tasked with finding the maximum load when the crane is at an angle of $35.8^\circ$ with the horizontal. Here, the torque formula remains the same, but now the angle is different, and we are solving for the force $F$. The equation becomes: $F = \frac{\tau}{L \cdot \sin(\theta)}$ where $\tau$ is the previously calculated maximum torque.

Let's compute the values for both parts. math

Given values

L = 19.0 # length of the crane arm in meters theta1_deg = 13.2 # angle in degrees for part 1 F_max = 706 # maximum load in newtons

Convert angle from degrees to radians

theta1_rad = math.radians(theta1_deg)

Calculate the torque for part 1

torque = F_max * L * math.sin(theta1_rad)

Now, for part 2, the angle is 35.8 degrees

theta2_deg = 35.8 theta2_rad = math.radians(theta2_deg)

Calculate the maximum load for part 2

F_max2 = torque / (L * math.sin(theta2_rad))

torque, F_max2### Part 1: The maximum torque that the crane can withstand with the maximum load of 706 N is approximately: $\tau = 3063.1 \, \text{N} \cdot \text{m}$

Part 2:

The maximum load the crane can handle at an angle of 35.8° with the horizontal is approximately: $F_{\text{max}} = 275.6 \, \text{N}$

Would you like further details on how these calculations were made, or do you have any other questions? Here are some related questions for you:

1. How does the angle between the crane arm and the horizontal affect the torque?
2. How can we derive the torque formula using basic principles of physics?
3. How would the torque change if the length of the crane arm were altered?
4. Why is torque important when determining the stability of structures like cranes?
5. What other forces or considerations might affect the crane's ability to lift loads?

Tip: Always ensure the angle is converted to radians when working with trigonometric functions in physics problems!