Rotational Mechanics: Comparing Cylinder P & Q

In summary, two solid cylinders, P and Q, with the same mass and radius, roll down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface while cylinder Q has most of its mass near the axis. The moment of inertia of P is greater than that of Q due to the distribution of mass. The instantaneous axis of rotation will depend on the position of the cylinders as they roll down the incline. Both cylinders will reach the ground with the same translational kinetic energy.
  • #1
joker94
3
0
Two solid cylinders p and Q of same mass and same radius start rolling down a fixed inclinded plane from the same height at the asme time.cylinder P has most of its mass concentrated near its surface while Q HAS MOST of its mass near the axis...
THEN
1)WHICH HAS MORE MOMENT OF INERTIA AND WHY?
2)WHAT IS THE INSTANTEOUS AXIS OF ROTATION??
3)DOES BOTH CYLINDERS REACH THE GROUND WITH SAME TRANSLATIONAL KINECTIC ENERGY?
 
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  • #2
welcome to pf!

hi joker94! welcome to pf! :wink:

tell us what you think, and then we'll comment! :smile:​
 
  • #3
hi...can we consider P as a hollow cylinder then moment of inertia about the axis(which is passing through its centre) would be MR^2...AND THEN WE CAN CONSIDER Q AS A SOLID CYLINDER HAVING MOMENT OF INERTIA MR^2/2 (^2 MEANS SQUARE AND / DIVIDE)...SO IT MEANS MOMENT OF INERTIA OF P IS MORE THAN MOMENT OF INERTIA OF Q.!
AM I CORRECT?
 
  • #4
joker94 said:
hi...can we consider P as a hollow cylinder then moment of inertia about the axis(which is passing through its centre) would be MR^2...AND THEN WE CAN CONSIDER Q AS A SOLID CYLINDER HAVING MOMENT OF INERTIA MR^2/2 (^2 MEANS SQUARE AND / DIVIDE)...SO IT MEANS MOMENT OF INERTIA OF P IS MORE THAN MOMENT OF INERTIA OF Q.!
AM I CORRECT?

Right answer, though of course it would be better not to have to assume the extreme case when the original question was posed more generally.
How about your questions 2 and 3?
 
  • #5


1) Cylinder Q has more moment of inertia because it has a greater distribution of mass towards its axis, which increases its resistance to rotational motion.
2) The instantaneous axis of rotation for both cylinders will be at the point of contact between the cylinder and the inclined plane.
3) No, the two cylinders will not reach the ground with the same translational kinetic energy. This is because the distribution of mass affects their rotational motion, and therefore their final velocity and kinetic energy. Cylinder Q, with its more concentrated mass towards the axis, will have a lower translational kinetic energy compared to cylinder P, which has more mass distributed towards its surface.
 

Related to Rotational Mechanics: Comparing Cylinder P & Q

1. What is rotational inertia and how does it differ between cylinders P and Q?

Rotational inertia is a measure of an object's resistance to changes in its rotational motion. It is determined by the mass and distribution of the object's mass. In the case of cylinders P and Q, the rotational inertia differs because of their respective shapes. Cylinder P has a solid, uniform shape, while cylinder Q has a hollow, ring-like shape. This difference in shape results in different distributions of mass, leading to different rotational inertias.

2. How does the radius of a cylinder affect its rotational motion?

The radius of a cylinder affects its rotational motion because it impacts the distribution of mass. A cylinder with a larger radius will have more mass concentrated farther from its axis of rotation, resulting in a higher rotational inertia. This means that a larger force will be required to change the rotational motion of a cylinder with a larger radius compared to one with a smaller radius.

3. What is the relationship between torque and rotational motion?

Torque is the force that causes an object to rotate. In rotational mechanics, it is directly related to an object's rotational motion. The greater the torque applied to an object, the greater the acceleration of its rotational motion. Additionally, the direction of the torque determines the direction of the object's rotational motion.

4. How does the density of a material affect the rotational inertia of a cylinder?

The density of a material does not directly affect the rotational inertia of a cylinder. However, it does impact the distribution of mass within the object. A cylinder made of a denser material will have more mass concentrated in a smaller space, resulting in a higher rotational inertia compared to a cylinder made of a less dense material with the same mass.

5. Can the rotational motion of cylinders P and Q be compared using their moment of inertia?

Yes, the moment of inertia is another term for rotational inertia and can be used to compare the rotational motions of cylinders P and Q. The moment of inertia takes into account the mass and distribution of mass of an object, making it a useful measure for comparing the rotational motion of different objects.

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