small ball of mass m is suspended by a string of length L. The string makes an angle β with the vertical. The ball revolves in a circle with an unknown constant angular speed w. The orbital plane of the ball is at a height h above the ground. Let g be the gravitational constant. You may ignore air resistance and the size of the ball. Later, the ball detaches from the string just as it passes the x-axis. It flies through the air and hits the ground at an unknown horizontal distance d from the point at which it detached from the string.
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
A small ball of mass m is suspended by a string of length L. The string makes an angle β with the vertical. The ball revolves in a circle with an unknown constant angular speed w. The orbital plane of the ball is at a height h above the ground. Let g be the gravitational constant. You may ignore air resistance and the size of the ball.
Later, the ball detaches from the string just as it passes the x-axis. It flies through the air and hits the ground at an unknown horizontal distance d from the point at which it detached from the string.
- What horizontal distance d does the ball traverse before it hits the ground? Express you answer in terms of some or all of the following: β, L, h
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