Question

Suppose that we repeat the experiment in the video but use hand-held weights of different masses and different initial speeds. In each of the cases below, the demonstrator starts with her arms extended (rotating at angular speed ?0) and then pulls her arms in (rotating at angular speed ?f). Rank the cases below based on their final angular speed ?f (that is, when the demonstrator’s arms are pulled in), from slowest to fastest. Since most of the mass of the demonstrator sits near the axis of rotation, ignore her mass for this question.

Rank the cases shown below from slowest final angular speed to fastest final angular speed. Here “final” refers to the angular speed when the demonstrator’s arms are pulled in. If two cases have the same angular speed, place one on top of the other.

https://mediaplayer.pearsoncmg.com/assets/secs-vtd49_consam

Hint 1.

Use conservation of angular momentum to relate the initial angular speed to the final angular speed. In this case, the moment of inertia of the system is I=2mr2, where m is the mass of a single hand-held weight and r is the distance between the rotation axis and the weight.

This is the original set.

I use L=Iw(omega), I=2mr^2,

L = 2mr^2w and get the new order. Kindly please check it.

Answer 1

Concepts and reason

The concept required to solve the given problem is law of conservation of angular momentum.

Rank the final angular speed with the help of conservation of angular momentum.

Fundamentals

The expression of the magnitude of the rotational angular momentum of the spinning disk is given as follows:

 

Here, I is the moment of inertial and  is the angular speed for one complete rotation.

The direction of angular momentum is given using the Right-hand rule. Curl the fingers in the direction of rotation then the thumb of your right hand will give the direction of angular momentum.

Conservation of Angular Momentum: It states that when no external torque acts on an object, the change in angular momentum remains constant. Mathematically, the law may be stated as,

Here, is the initial angular momentum and is the final angular momentum.

The moment of inertia of an object is given as,

Here, is the mass and is the radius of gyration.

According to the law pf conservation of angular momentum,

…… (1)

Here, is the initial and final angular momentum.

The angular momentum is given by,

…… (2)

Here, is the position vector and is the linear momentum.

Also, momentum is given by,

Here, is the mass and is the velocity.

Substitute for in equation (2).

Here, is the angle between the position vector and velocity vector.

Thus, equation (1) becomes,

…… (3)

Since the position vector is perpendicular to the velocity vector, the angle will be .

Also, angular speed is given by,

Substitute for and for in equation (3).

Case 1: initial angular speed, mass

The final angular speed will be,

Case 2: initial angular speed , mass

The final angular speed will be,

Case 3: initial angular speed , mass

The final angular speed will be,

Case 4: initial angular speed , mass

The final angular speed will be,

Case 5: initial angular speed , mass

The final angular speed will be,

Case 6: initial angular speed , mass

The final angular speed will be,

Since the ratio of radius remains same for all the cases, the final angular speed will only depend on the initial angular speed.

The final angular speed will be the slowest for the case with initial angular speed , mass and initial angular speed , mass and the fastest for the case with initial angular speed , mass and initial angular speed, mass.

The ranking of the cases for the different cases given on the basis of angular speed is,

Ans:

The ranking of the cases for the different cases given on the basis of angular speed is,

Answer 2

Answer #1: Concepts and Reason is correct, but the way they listed their solution was kind of confusing to me so here is a screenshot just in case anyone else needs it.