A 3.0 x 102 g solid, uniform sphere of radius 1.5 cm is released from rest at the top of a ramp. When at the top of the ramp, the sphere’s center of mass is 5 m from the ground. It rolls without slipping down the ramp and enters a vertical loop-de-loop. At the top of the loop, the sphere’s center of mass is 2.2 m from the ground. Ignore air resistance in this problem. a) Using energy considerations, what is the sphere’s translational velocity at the top of the loop? b) At the top of the loop, what is the magnitude of the sphere’s angular momentum about its center of mass? c) Consider the loop as a perfect circle with radius 1.1 m. When the sphere is at the top of the loop, what is the magnitude of the sphere’s angular momentum about the center of the loop?
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 3.0 x 102 g solid, uniform sphere of radius 1.5 cm is released from rest at the top of a ramp. When at the top of the ramp, the sphere’s center of mass is 5 m from the ground. It rolls without slipping down the ramp and enters a vertical loop-de-loop. At the top of the loop, the sphere’s center of mass is 2.2 m from the ground. Ignore air resistance in this problem.
a) Using energy considerations, what is the sphere’s translational velocity at the top of the loop?
b) At the top of the loop, what is the magnitude of the sphere’s
c) Consider the loop as a perfect circle with radius 1.1 m. When the sphere is at the top of the loop, what is the magnitude of the sphere’s angular momentum about the center of the loop?
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