Dynamics angular velocity question

[JGreen48]

[SOLVED] dynamics angular velocity question

I've tried this problem with a simple way of kinetic energy and potential, but I am now where close to the answer of 11.1. There must be a way of doing it I don't know of.

The .1 kg slender bar and .2 kg cylindrical disk are released from rest with the bar horizontal. The disk rolls on the curved surface. What is the angular velocity when the bar is vertical. The radius of the disk is 40 mm. and the length of the bar is 120 mm.

A quick drawing of it is attahced.

 

[Valkarie]

The solution to this problem can be found by using the conservation of energy. Since the system is initially at rest, the total energy of the system is zero. When the bar is vertical, the gravitational potential energy of the system will be at a maximum, so the total energy of the system will still be zero. Since the angular velocity of the disk is increasing, the kinetic energy of the system must also increase. The kinetic energy of the system will be equal to the change in gravitational potential energy. Since the mass of the bar and the disk are known, the change in potential energy can easily be calculated by multiplying the mass by the height. The angular velocity of the disk can be found by using the equation for rotational kinetic energy, which is:K = 1/2 I ω2 Where I is the moment of inertia of the disk and ω is the angular velocity. The moment of inertia of the disk can be calculated by using the equation for a solid cylinder, which is: I = 1/2 mr2 Where m is the mass of the disk and r is the radius. Putting all these equations together, we get: K = 1/2 (1/2 mr2) ω2 Substituting the values for m, r, and the change in potential energy, we get: ω2 = 2(0.2)(0.04)2/ 0.2(9.81)(0.12) Solving for ω, we get an angular velocity of 11.1 rad/s.

 

[piercas]

It seems like you are on the right track with using kinetic energy and potential energy to solve this problem. However, there are a few additional considerations that need to be taken into account in order to arrive at the correct answer of 11.1.

Firstly, it is important to note that the bar and disk will have different moments of inertia, which will affect their rotational motion. The moment of inertia for a slender bar is 1/12 * m * L^2, where m is the mass and L is the length. The moment of inertia for a cylindrical disk is 1/2 * m * R^2, where m is the mass and R is the radius. Using these equations, we can calculate the moments of inertia for the bar and disk in this problem.

Next, we need to consider the conservation of energy. At the initial position, the bar and disk have potential energy due to their height above the ground. As they roll down the curved surface, this potential energy is converted into kinetic energy. At the final position when the bar is vertical, all of the potential energy has been converted to kinetic energy.

Finally, we can use the equation for rotational kinetic energy, 1/2 * I * w^2, where I is the moment of inertia and w is the angular velocity, to solve for w. Setting the initial potential energy equal to the final kinetic energy, we can solve for w and arrive at the answer of 11.1.

In summary, the key to solving this problem is to consider the different moments of inertia, conservation of energy, and the equation for rotational kinetic energy. With these factors in mind, we can arrive at the correct answer for the angular velocity of the system.