A solid cylinder with a mass of 300.0g and a radius of 20.0cm is rotating at 50.0 rad/s. A metal hoop of mass 200.0g and radius 10.0cm, initially not rotating, is dropped onto the rotating solid cylinder so that their rotational axes align. Determine the angular speed of the 2 objects as they rotate together. Report your answer in radians per second, but only enter in the numerical value.
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.
Given quantities:
mass of cylinder (M) = 0.3 kg
radius of cylinder (R) = 0.2 m
angular speed (w1) = 50 rad/s
mass of metal hoop (m) = 0.2 kg
radius of metal hoop (r) = 0.1 m
Let, the common angular speed with which the two objects rotate be 'W' , moment of inertia be denoted as 'I'
As per question, it is clear that there is no external torque acting on the system. So, the angular momentum will remain conserved about the axis of rotation
First calculate the initial angular momentum of the system as follows:
Li = Iw1 = ( MR2w1 / 2 ) ..... ( as I = MR2/ 2 )
= ( 0.30.22 / 2 )
= 0.3 kg.m2/s
Similarly, the final angular momentum of the system will be :
Lf = ( MR2/2 + mr2)W ....( as two objects move with same angular speed )
= [ 0.32/ 2 + 0.2(0.1)2 ]W
= (0.008)W
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