Problem 26 - University Physics with Modern Physics

Question

Does a particle traveling in a straight line have an angular momentum? Explain.

Step-by-Step Solution

Answer

Answer: Yes, a particle traveling in a straight line does have angular momentum. Its magnitude depends on the perpendicular distance from the reference point and the linear momentum of the particle.

Step 1: Define Angular Momentum

Angular momentum is a property associated with the rotation of any object, or system of objects, about a specific reference point. We use the letter 'L' to represent it, and it is defined as the cross product of the distance vector 'r' (the vector from the reference point to the position of the particle) and the linear momentum 'p' of the particle. Mathematically, L = r x p.

Step 2: Establish the situation

We have a particle traveling in a straight line. Let's assume this line is parallel to the x-axis. We will consider a reference point on the y-axis, at a perpendicular distance d from the particle's path.

Step 3: Determine the linear momentum and position vector

Since the particle is moving in a straight line along the x-axis, its linear momentum, p, will be in the x-direction. Let's denote its magnitude as m*v, where m represents the mass of the particle, and 'v' is its constant velocity. The position vector, r, will be the vector from the reference point to the particle's position. As the particle moves in a straight line parallel to the x-axis, its position vector will have components in both the x and y directions. The y-component will be equal to the perpendicular distance d from the reference point to the particle's path.

Step 4: Calculate the cross product of r and p

To find the angular momentum, we need to calculate the cross product of the position vector r and the linear momentum p. The cross product of two vectors in three-dimensional space can be given by: L = r x p = (r_y * p_z - r_z * p_y) i + (r_z * p_x - r_x * p_z) j + (r_x * p_y - r_y * p_x) k Since the particle is moving in a straight line parallel to the x-axis, the linear momentum has no component in the y or z directions, so p_y = p_z = 0. The position vector r is confined to the xy-plane, so the z-component of r will also be zero i.e., r_z = 0. Given this information, we can now simplify the cross product L = (r_y * 0 - 0 * 0) i + (0 * p_x - r_x * 0) j + (r_x * 0 - r_y * p_x) k

Step 5: Evaluate the cross product and determine angular momentum

After simplifying the expression for the cross product, we obtain: L = - r_y * p_x k Since r_y = d (the perpendicular distance from the reference point to the particle's path), we can write the angular momentum as: L = - d * p_x k Since d and p_x are both non-zero, we can conclude that the particle traveling in a straight line does have an angular momentum. The negative sign indicates that the angular momentum is in the opposite direction of the z-axis, and its magnitude depends on the perpendicular distance d from the reference point and the linear momentum of the particle.