A PARTICLE OF MASS M IS SHOT WITH an initial velocity v making an angle b with the horizontal,the particle moves in the gravitational field of the earth. find the angular momentum of the particle about the origin when the particle is at the origin ,at the highest point of its trajectory, just before it hits the ground and what torque causes its angular momentum to change?
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To find the angular momentum of a particle, you need to know both its linear momentum and the distance from the origin at which the angular momentum is being measured. In this case, the angular momentum about the origin can be calculated at three different positions: when the particle is at the origin, at the highest point of its trajectory, and just before it hits the ground.
1. Angular Momentum at the Origin:
When the particle is at the origin, its linear momentum is zero since it is not moving. Therefore, the angular momentum about the origin is also zero because the distance from the origin is zero.
Angular Momentum about the Origin (at the origin) = 0
2. Angular Momentum at the Highest Point of the Trajectory:
At the highest point of the trajectory, the particle's vertical velocity becomes zero, but its horizontal velocity is still present. The angular momentum about the origin is determined by the linear momentum and the distance from the origin.
First, we find the linear momentum of the particle. The linear momentum (p) can be calculated by multiplying the mass (m) of the particle by its velocity (v) at the highest point of the trajectory.
Linear Momentum at the Highest Point = m * v
The distance from the origin is the vertical component of the particle's position at the highest point, which is given by:
Distance from the Origin = vertical height (h)
The angular momentum is the product of the linear momentum and the distance from the origin:
Angular Momentum about the Origin (at the highest point) = (m * v) * h
3. Angular Momentum just before it Hits the Ground:
Just before the particle hits the ground, both its horizontal and vertical velocities become non-zero. Again, the angular momentum about the origin is determined by the linear momentum and the distance from the origin.
The linear momentum (p) can be calculated as before:
Linear Momentum just before hitting the ground = m * v
The distance from the origin is the horizontal component of the particle's position just before hitting the ground, which is given by:
Distance from the Origin = horizontal displacement (d)
The angular momentum is the product of the linear momentum and the distance from the origin:
Angular Momentum about the Origin (just before hitting the ground) = (m * v) * d
The torque causing the angular momentum to change in this scenario is the gravitational torque. When the particle is displaced from the origin, the force of gravity on the particle produces a torque about the origin. Torque is the rate of change of angular momentum and is given by:
Torque = dL/dt
In this case, the torque causing the angular momentum to change is the gravitational torque acting on the particle due to Earth's gravity. The gravitational torque is calculated as the cross product of the position vector and the gravitational force vector:
Gravitational Torque = r x F
However, since the gravitational force is acting along the vertical direction (opposite to the position vector), the torque due to gravity will be zero. Therefore, no torque is causing the angular momentum to change in this scenario.