Question

a particle of mass m is projected with speed u at an angle θ with horizontal from the ground find angular momentum about point of projection when particle again at same horizontal level

Answer 1

The angular momentum of a particle projected at an angle with the horizontal can be calculated by multiplying its mass, horizontal range, initial velocity, and the cosine of the projection angle, considering the direction given by the right-hand rule.

Step-by-step explanation:

The question involves calculating the angular momentum of a particle projected at an angle θ with initial velocity u from the ground, when the particle is again at the same horizontal level. The angular momentum (ℒ) about the point of projection can be found using the formula ℒ = r × p, where r is the position vector from the origin to the particle, and p is the linear momentum of the particle. At the same horizontal level, r is the horizontal range of the projectile, and the linear momentum p has the direction and magnitude of the projectile's velocity. As the particle's vertical velocity component is zero at the same horizontal level, p can be assumed to be only in the horizontal direction.

To calculate the angular momentum specifically:

1. Calculate the horizontal range (R) of the projectile using R = (u^2 × sin(2θ)) / g, where g is the acceleration due to gravity.
2. The horizontal component of the velocity remains constant, so we use v_x = u × cos(θ).
3. The linear momentum p = m × v_x.
4. Since r and p are perpendicular at the point where the particle returns to the same horizontal level, ℒ = |r| × |p|.
5. Therefore, ℒ = m × R × u × cos(θ).

The direction of the angular momentum is determined by the right-hand rule.