Why can't I treat the disk as a point mass?

In summary, the disk rotated about its center. However, if you treat the disk as a point mass, you get a different answer.
  • #1
cory21
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0
Homework Statement
A grandfatherโ€™s clock consists of a disk of mass, ๐‘ด attached to the end of rod of negligible mass and length, ๐‘ณ. The grandfatherโ€™s clock is hanging vertically initially from a hinge at point ๐‘จ.
A lump of clay mass, ๐’Ž moving horizontally at speed, ๐’— collides with and sticks to the center of the disk, causing the grandfatherโ€™s clock to rise to a maximum angle ๐œฝ๐’Ž๐’‚๐’™.

Which of the following is an expression for the angular speed angular speed, ๐Ž of the grandfather's clock (with a lump of clay sticking to center of disk) just after the collision? [Note: Assume that the clay is a point mass. Moment of inertia of disk about axis through center of disk I_disk=1/2MR^2]
Relevant Equations
๐ฟ๐‘ก๐‘œ๐‘ก,๐‘ƒ = ๐ฟ๐‘Ÿ๐‘œ๐‘ก๐ถ๐‘€,๐‘Ÿ๐‘œ๐‘‘ + ๐ฟ๐‘Ÿ๐‘œ๐‘ก๐ถ๐‘€,๐‘‘๐‘–๐‘ ๐‘˜ + ๐ฟ๐‘ก๐‘Ÿ๐‘Ž๐‘›๐‘ ,๐‘ƒ,๐‘Ÿ๐‘œ๐‘‘ + ๐ฟ๐‘ก๐‘Ÿ๐‘Ž๐‘›๐‘ ,๐‘ƒ,๐‘‘๐‘–๐‘ ๐‘˜
Screen Shot 2022-11-24 at 15.20.55.png


Since the question made no indication of the disk rotating about its center, I just straight up assumed that the disk did not rotate about its center, and instead treated it as a point mass. However, to my surprise my calculations did not bear me any fruit. Below is my first attempt at the solution, where you could clearly see that my calculations did not yield me an answer that's even remotely close as to what was offered in the MCQ.

Screen Shot 2022-11-24 at 15.26.42.png


If I treated the disk to have rotated around its center however, I would get D as an answer. Below is my second attempt at the solution

Screen Shot 2022-11-24 at 15.32.26.png


So, am I to assume that the disk experienced some translational motion along with rotational motion about the disk's center of mass, because the question along with the diagram isn't very clear.
 
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  • #2
cory21 said:
So, am I to assume that the disk experienced some translational motion along with rotational motion about the disk's center of mass, because the question along with the diagram isn't very clear.
Yes. I think you are supposed to assume that the disk is attached to the rod so that the disk doesn't rotate relative to the rod.

Suppose you paint an orange line on the disk.

1669323120416.png

You can see how the disk rotates through some angle ##\phi## as the rod swings through an angle ##\theta##. What is the relation between ##\theta## and ##\phi##?
 
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  • #3
Here is a figure that I posted in another thread where the same issue arose. The Moon showing the same side to the Earth is an illustration of this idea of one spin revolution per orbit revolution.

PendulumDisk.png
 
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  • #4
What surprised me about this scenario is that having the bob attached rigidly to the shaft, rather than on a free axle, increases the energy loss in the impact.
 

Related to Why can't I treat the disk as a point mass?

Why can't I treat the disk as a point mass?

1. What is the difference between a disk and a point mass?

A disk is a three-dimensional object with a finite thickness, while a point mass is a theoretical object with no dimensions. This means that a disk has mass distributed throughout its volume, while a point mass has all its mass concentrated at a single point.

2. Can't I just use the center of mass of the disk as a point mass?

The center of mass of a disk is a point that represents the average position of all the mass in the disk. However, it does not accurately represent the distribution of mass within the disk. Treating the disk as a point mass ignores the effects of the mass distribution on its motion.

3. Why is it important to consider the mass distribution of an object?

The mass distribution of an object affects its moment of inertia, which is a measure of its resistance to rotation. Treating a disk as a point mass would result in an incorrect moment of inertia, leading to inaccurate predictions of its rotational motion.

4. Can't I just use the mass of the disk and its radius to calculate its motion?

No, because the mass and radius alone do not fully describe the distribution of mass within the disk. The thickness and density of the disk also play a crucial role in determining its moment of inertia and rotational motion.

5. Is there ever a situation where I can treat a disk as a point mass?

In some cases, if the thickness of the disk is negligible compared to its radius, it can be treated as a point mass. However, this is only an approximation and may not accurately represent the true behavior of the disk.

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