Rotational Dynamics of 2 Particles and 2 Rods

In summary, in the given scenario, two particles with mass m = 2.30 kg each are connected to each other and to a rotation axis at O by two thin rods with length d = 0.670 m and mass M = 0.360 kg. The combination is rotating around the rotation axis with an angular speed of ω = 0.211 rad/s. To find the total rotational inertia, the parallel axis theorem should be used for the rods, and the contributions from the masses should be added. The second rod, which is not touching the axis, will need to be computed separately.
  • #1
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In Fig. 10-35, two particles, each with mass m = 2.30 kg, are fastened to each other, and to a rotation axis at O, by two thin rods, each with length d = 0.670 m and mass M = 0.360 kg. The combination rotates around the rotation axis with angular speed ω = 0.211 rad/s. Measured about O, what are the combination's (a) rotational inertia and (b) kinetic energy?
(I'm not sure how to post pictures on here, because the picture is on wileyplus, and to view it, you need a password)

But for this problem, I am not even sure where to start. I know that I need to find the total inertia, but how many parts should i split it into? Should I take each rod seperately and each particle seperately, and use the summation formula for inertia, or should i use the parallel axis theorem? I am just so lost on where to start.
 
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  • #2
If the configuration is axis-rod-particle-rod-particle, then you will want to use the parallel axis theorem for the rods and add the contributions from the masses. The parallel axis theorm has already been used for the rod rotating at one end in most places where you look up moments of inertia. Since the second rod is not touching the axis, you will have to compute it yourself.
 
  • #3


As a scientist, my response would be to approach this problem by breaking it down into smaller, manageable parts. In this case, since we are dealing with rotational dynamics, it would be helpful to consider the individual rotational inertia of each component (particles and rods) and then combine them to find the total rotational inertia of the system.

To find the rotational inertia of each particle, we can use the formula I = mr^2, where m is the mass and r is the distance from the rotation axis. In this case, both particles have the same mass (m = 2.30 kg) and are located at a distance of d/2 = 0.335 m from the rotation axis. Therefore, the rotational inertia of each particle is I = (2.30 kg)(0.335 m)^2 = 0.387 kg*m^2.

Next, we can use the parallel axis theorem to find the rotational inertia of each rod. This theorem states that the rotational inertia of an object about an axis parallel to its center of mass is equal to the rotational inertia about its center of mass plus the product of its mass and the square of the distance between the two axes. In this case, the center of mass of each rod is located at its midpoint, so the distance between the rotation axis and the center of mass is d/2 = 0.335 m. Therefore, the rotational inertia of each rod is I = (1/12)(M)(d^2) + (M)(0.335 m)^2 = 0.024 kg*m^2.

To find the total rotational inertia of the system, we can simply add the individual inertias of the particles and rods. So, the total rotational inertia of the system is I = (2)(0.387 kg*m^2) + (2)(0.024 kg*m^2) = 0.846 kg*m^2.

To find the kinetic energy of the system, we can use the formula KE = (1/2)(I)(ω^2), where I is the rotational inertia and ω is the angular speed. Plugging in the values, we get KE = (1/2)(0.846 kg*m^2)(0.211 rad/s)^2 = 0.0090 J.

In summary, to solve this problem, we used the formulas for rotational inertia and kinetic energy, as well as the parallel axis theorem, to find the
 

1. What is rotational dynamics?

Rotational dynamics is a branch of physics that deals with the motion of objects that rotate or spin around a fixed axis. It involves studying the forces and torques that act on these objects, as well as their angular velocities and accelerations.

2. How are rotational dynamics and linear dynamics related?

Rotational dynamics and linear dynamics are both branches of classical mechanics that describe the motion of objects. Rotational dynamics is concerned with the motion of objects that rotate, while linear dynamics deals with the motion of objects that move in a straight line. They are related through the concepts of force, mass, and acceleration, as well as the laws of motion.

3. What are the equations of rotational dynamics?

The main equations of rotational dynamics are the moment of inertia, torque, and angular acceleration. These equations describe the relationship between the angular velocity and acceleration of an object and the forces and torques acting on it.

4. How do you calculate the moment of inertia?

The moment of inertia is a measure of an object's resistance to rotation and depends on the object's mass and its distribution around the axis of rotation. It can be calculated using the formula I = Σmr², where m is the mass of each particle and r is its distance from the axis of rotation.

5. Can you give an example of rotational dynamics in real life?

One example of rotational dynamics in real life is the motion of a spinning top. As the top spins, it experiences a torque due to gravity, causing it to precess and eventually topple over. The moment of inertia of the top and the forces acting on it determine its motion and how long it will continue spinning before falling over.

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