To tackle the problem involving two forces acting on a rigid body, we need to break it down into manageable steps. Here, we will first derive an expression for the resultant torque and then apply the given values to find its magnitude and direction.

Understanding Torque

Torque (\( \tau \)) is a measure of how much a force acting on an object causes that object to rotate. It depends on three factors: the magnitude of the force applied, the distance from the pivot point (also known as the lever arm), and the angle at which the force is applied. The mathematical representation of torque is given by:

• \( \tau = r \cdot F \cdot \sin(\theta) \)

Here, \( r \) is the distance from the pivot point to the point of application of the force, \( F \) is the magnitude of the force, and \( \theta \) is the angle between the force vector and the lever arm.

Calculating Resultant Torque

For multiple forces acting on the same point, the total torque is the vector sum of the individual torques. If we have two forces \( F_1 \) and \( F_2 \) applied at distances \( r_1 \) and \( r_2 \) from the pivot, the resultant torque can be expressed as:

• \( \tau_{\text{resultant}} = \tau_1 + \tau_2 \)

Substituting for each torque, we get:

• \( \tau_{\text{resultant}} = r_1 \cdot F_1 \cdot \sin(\theta_1) + r_2 \cdot F_2 \cdot \sin(\theta_2) \)

Plugging in the Values

Now that we have our expression for the resultant torque, we can substitute the given values:

• \( r_1 = 1.30 \, \text{m} \)
• \( r_2 = 2.15 \, \text{m} \)
• \( F_1 = 4.20 \, \text{N} \)
• \( F_2 = 4.90 \, \text{N} \)
• \( \theta_1 = 75.0^\circ \)
• \( \theta_2 = 58.0^\circ \)

Calculating the individual torques:

• \( \tau_1 = 1.30 \cdot 4.20 \cdot \sin(75^\circ) \)
• \( \tau_2 = 2.15 \cdot 4.90 \cdot \sin(58^\circ) \)

First, we need to find the sine values:

• \( \sin(75^\circ) \approx 0.9659 \)
• \( \sin(58^\circ) \approx 0.8480 \)

Now we can compute:

• \( \tau_1 = 1.30 \cdot 4.20 \cdot 0.9659 \approx 5.245 \, \text{N}\cdot\text{m} \)
• \( \tau_2 = 2.15 \cdot 4.90 \cdot 0.8480 \approx 9.042 \, \text{N}\cdot\text{m} \)

Adding these two torques together to find the resultant torque:

• \( \tau_{\text{resultant}} = 5.245 + 9.042 \approx 14.287 \, \text{N}\cdot\text{m} \)

Direction of the Resultant Torque

The direction of the torque can be determined by the right-hand rule. If you curl the fingers of your right hand in the direction of the force's rotation, your thumb points in the direction of the torque vector. Here, since both forces are applied at angles that contribute positively to the rotation, the resultant torque will also tend to rotate the object in the same direction.

In summary, the magnitude of the resultant torque acting on the body is approximately 14.29 N·m, and the direction can be found using the right-hand rule, indicating a positive rotation about the pivot point. This analysis illustrates not only how to calculate torque effectively but also the physical implications of the forces acting on the rigid body.