A particle of mass m is projected with a velocity v making...

A particle of mass $m$ is projected with a velocity $v$ making an angle of $30^{\circ}$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $h$ is (a) $\frac{\sqrt{3}}{2} \frac{m v^{2}}{g}$ (b) zero (c) $\frac{m v^{3}}{\sqrt{2} g}$ (d) $\frac{\sqrt{3}}{16} \frac{m v^{3}}{g}$

A particle of mass $m$ is projected with a velocity $v$ making an angle of $30^{\circ}$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $h$ is
(a) $\frac{\sqrt{3}}{2} \frac{m v^{2}}{g}$
(b) zero
(c) $\frac{m v^{3}}{\sqrt{2} g}$
(d) $\frac{\sqrt{3}}{16} \frac{m v^{3}}{g}$

Key Concepts

Cross Product in Mechanics

The cross product in mechanics is a mathematical operation used to determine the angular momentum of a particle. It allows the calculation of the vector quantity perpendicular to both the position and momentum vectors, and its magnitude captures the rotational effect with respect to a specific point.

Vector Decomposition

In analyzing projectile motion, vector decomposition is used to break the initial velocity into horizontal and vertical components. This is crucial for determining the particle's position and velocity at any point along its trajectory, such as at the maximum height where the vertical component becomes zero.

Angular Momentum

Angular momentum is a measure of the amount of rotation a particle has about a chosen point, and is defined as the cross product of the position vector and the linear momentum vector (L = r × p). Its magnitude depends on the distance from the point, the particle's mass and speed, and the sine of the angle between the position and velocity vectors.

Projectile Motion

Projectile motion describes the motion of an object that is launched into the air subject only to the acceleration due to gravity. This concept involves analyzing both horizontal and vertical components of motion separately, where the horizontal velocity remains constant and the vertical velocity is affected by gravity.

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