A cylindrical pole is inserted into a frozen lake so the...

A cylindrical pole is inserted into a frozen lake so the pole stands vertically. One end of a rope is attached to a point on the surface of the pole near where it enters the ice, and the rope is then laid out in a straight line on the 12 Rigid-Body Dynamics surface. An ice skater with initial velocity $v_0$ approaches the opposite end of the rope, moving perpendicular to the rope. As she reaches the rope she grabs it and holds on, so the rope winds up around the pole. (a) When the rope is half wound up, what is her speed, assuming there is no friction between the rope or herself and the ice? (b) Has there been any change in her kinetic energy? If so, identify what positive or negative work has been done on her, and by what source. (c) Has there been any change in her angular momentum? If so, identify the source of the torque done upon her.

Key Concepts

Work Done by Constraint Forces

Constraint forces, which limit the motion of a system (like the tension in the rope that forces the skater’s path to change as the rope winds), can do work on the system. This work can change the kinetic energy even in the absence of friction, thus illustrating how forces that do not perform work in idealized holonomic constraints may become sources of energy change when the constraints themselves are dynamic.

Geometric Constraints in Rigid-Body Dynamics

The geometry of the system, particularly the winding of a rope around a cylindrical pole, imposes non-trivial constraints on motion. These geometric constraints link radial and angular movements, leading to changes in the path and speed of the moving body. A clear understanding of how these constraints affect the dynamics is fundamental when analyzing the evolution of the system’s motion.

Conservation of Energy

In a frictionless system, the total mechanical energy remains constant unless external work is done. However, even without friction, forces such as tension in the rope can do work on a moving object, changing its kinetic energy. Understanding energy conservation is crucial in analyzing how the skater’s speed changes as the rope winds around the pole.

Angular Momentum and Torque

Angular momentum is conserved in an isolated system unless an external torque is applied. In problems involving curved motion or changing constraints, forces (like the tension pulling the skater inward) can impart a torque, thereby altering the angular momentum of the skater. Recognizing the relationship between torque and angular momentum change is essential for studying such problems.

Transcript

00:01 In this question, given data is, that is there are two skaters is given whose mass are same, that is, mass that is m, is equal to 50 kg.

00:20 And now in the question it is saying that they are approaching each other along a parallel path separated by 3 meters.

00:29 So you can see that here i'm making a figure to explain the data.

00:33 So consider that this is a first skater and this is a second skater.

00:44 So they are approaching each other by a parallel path and separated by that is 3 meter.

00:54 In the question it is saying that and the velocity is given for both that is same in their opposite direction.

01:02 That is 1 .5, sorry, 1 .4 meter per second.

01:06 And for this second skater velocity is given that is same 1 .4 meter per second in this direction.

01:16 So now further in the question it is saying that the first skater is grabbing a end of pole.

01:27 So consider that this is the pole whose one end is grabbed by first skater and second end is grabbed by second skater.

01:36 And then after grabbing they are rotating around a axis which is passing through the center of this pole so this data is given and in the question it is saying that the friction between skate and ice is negligible so now in the question there are parts given so i will answering all the parts one by one so let's start answering this question so in the part a it is asking that we we have to find the radius of circle.

02:12 So my answer is for part a, that is, radius will be equal to looking this figure i consider it.

02:25 That is radius of circle.

02:29 When after rotating this pole by two skaters, then it will be equal to that is 3 over 2.

02:40 That means the distance between both separators, both skaters, sorry.

02:48 Both skaters that is 3 meter divided by 2 so simplify it then i will get the value of radius that is are that is equal to 1 .5 meter so this is the answer for part a now come to the next part so in the part b is it is asking that we have to find the angular speed of scatters so angular speed of skaters will be written as that is omega is equal to v over r, v is given that is equal, that is 1 .4 meter per second.

03:27 Fitted by r, r is that is 1 .5 meter.

03:31 So simplify it, then i will get the angular speed that is equal to 0 .93 radiance per second.

03:37 So by this angular speed, that is 0 .93 radian per second, the both skaters are rotating in the direction that is counterclockwise around the center of this pole.

03:50 So this is the answer for part b and this is the answer for part a.

03:58 Now come to the next part c so in the part c it is asking that we have to find the kinetic energy of two skaters system so that means for finding this kinetic energy first i will try to find total rotational inertia of system so it will be written as that is value will be m1 r1 square plus m2 r2 r2 square.

04:24 That means rotational inertia for first skater and rotational inertia for second skater...

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