Force F = ─10ĵ N is exerted on a particle at 𝓇 = (5î＋5ĵ) m. What is the torque on the particle about the origin?

Verified step by step guidance

1. Identify the position vector 𝓇 and the force vector F. In this case, 𝓇 = (5î + 5ĵ) m and F = ─10ĵ N.
2. Recall the formula for torque, τ, which is given by the cross product of the position vector 𝓇 and the force vector F: τ = 𝓇 × F.
3. Calculate the cross product. For vectors 𝓇 = (x₁î + y₁ĵ + z₁𝑘) and F = (x₂î + y₂ĵ + z₂𝑘), the cross product is calculated as: τ = (y₁z₂ - z₁y₂)î - (x₁z₂ - z₁x₂)ĵ + (x₁y₂ - y₁x₂)𝑘.
4. Substitute the components of 𝓇 and F into the cross product formula. Here, since the 𝑘 components of both vectors are zero, simplify the calculation focusing on the î and ĵ components.
5. Evaluate the simplified expression to find the components of the torque vector τ in terms of î, ĵ, and 𝑘.

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object, calculated as the cross product of the position vector and the force vector. It determines how effectively a force can cause an object to rotate about a pivot point, such as the origin in this case. The direction of torque is given by the right-hand rule, indicating the axis of rotation.

Cross Product

The cross product is a mathematical operation that takes two vectors and produces a third vector that is perpendicular to the plane formed by the original vectors. In the context of torque, the cross product of the position vector and the force vector yields the torque vector, which encapsulates both the magnitude and direction of the rotational effect of the force.

Position Vector

The position vector represents the location of a point in space relative to a reference point, typically the origin. In this problem, the position vector 𝓇 = (5î + 5ĵ) m indicates the particle's position in a two-dimensional Cartesian coordinate system. This vector is essential for calculating torque, as it defines the lever arm through which the force acts.