A metal bar is in the xy-plane with one end of the bar at the origin. A force F = (7.00 N)i + (-3.00 N)j is applied to the bar at the point x = 3.00 m, y = 4.00 m. (b) What are the magnitude and direction of the torque with respect to the origin produced by F?

Verified step by step guidance

1. First, understand that torque (τ) is a measure of the rotational force applied to an object. It is calculated using the cross product of the position vector (r) and the force vector (F). The formula for torque is τ = r × F.
2. Identify the position vector r from the origin to the point where the force is applied. In this case, r = (3.00 m)i + (4.00 m)j.
3. Write the force vector F, which is given as F = (97.00 N)i + (-3.00 N)j.
4. Calculate the cross product r × F. The cross product in two dimensions can be simplified using the determinant of a matrix: |i j k| |3.00 4.00 0| |97.00 -3.00 0|. Since the k-component is zero, the torque will be in the k-direction.
5. Determine the magnitude of the torque using the formula: |τ| = |r| |F| sin(θ), where θ is the angle between r and F. Alternatively, use the determinant method to find the magnitude directly from the cross product calculation.

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object. It is calculated as the cross product of the position vector (r) and the force vector (F), given by τ = r × F. The magnitude of torque is rFsinθ, where θ is the angle between r and F. Torque determines how effectively a force can rotate an object around a pivot point.

Cross Product

The cross product is a mathematical operation used to find a vector perpendicular to two given vectors in three-dimensional space. For vectors A and B, the cross product A × B results in a vector whose magnitude is |A||B|sinθ, where θ is the angle between A and B. The direction of the resulting vector is determined by the right-hand rule.

Right-Hand Rule

The right-hand rule is a mnemonic for determining the direction of the cross product vector. To apply it, point your right hand's fingers in the direction of the first vector (r), curl them towards the second vector (F), and your thumb will point in the direction of the resulting torque vector. This rule helps visualize the orientation of rotational effects in three-dimensional space.