Inertia of Disk and Block

In summary, the moment of inertia of the block about the rotation axis is 0.0018 kg*m^2 and the moment of inertia of the disk about the rotation axis is 0.7354 kg*m^2. When the system is rotating with an angular velocity of 4.2 rad/s, its energy is 3.0846 J. If the system has an angular acceleration of 8.1 rad/s^2 while rotating at 4.2 rad/s, the magnitude
  • #1
pleasehelpme6
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Homework Statement


A uniform disk of mass Mdisk = 4.9 kg and radius R = 0.2 m has a small block of mass mblock = 2 kg on its rim. It rotates about an axis a distance d = 0.17 m from its center intersecting the disk along the radius on which the block is situated.

a) What is the moment of inertia of the block about the rotation axis?

b) What is the moment of inertia of the disk about the rotation axis?

c) When the system is rotating about the axis with an angular velocity of 4.2 rad/s, what is its energy?

d) If while the system is rotating with angular velocity 4.2 rad/s it has an angular acceleration of 8.1 rad/s2, what is the magnitude of the acceleration of the block?

Homework Equations


I = mR^2


The Attempt at a Solution


I tried I = mR^2 for m = 4.9 and R = 0.2 for part b and also m = (4.9 + 2) but that was also not right.

For the block i tried I = (2)*(.17^2) but that didn't work either, so I'm completely lost.

Please help.
 
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  • #3
I found I = .5*m*R^2, and i tried both of the following:

I = .5*4.9*(.2^2) and
I = .5*4.9*(.2^2) + (2)(.17^2)

and neither worked.
 
  • #4
Any other suggestions?
 
  • #5
pleasehelpme6 said:
I found I = .5*m*R^2,
Good. That's the moment of inertia of a disk about its center. But its not rotating about its center, it's rotating about the specified point. So you'll need to make use of the parallel axis theorem to find the moment of inertia of the disk about the given axis.

and i tried both of the following:

I = .5*4.9*(.2^2) and
I = .5*4.9*(.2^2) + (2)(.17^2)

and neither worked.
Neither of those is quite right. The first is the moment of inertia of the disk about its center. The second is almost right--Look up the parallel axis theorem.

Note that you're working on (b). What about (a)? What's the moment of inertia of the block about the axis? (Treat the block as being a small mass.)
 
  • #6
You also need the parallel axis theorem since the disk is not rotating about its center.

http://hyperphysics.phy-astr.gsu.edu/hbase/parax.html#pax

I had to draw this out in order to really see it. The block is actually rotating around in a circle with a radius of 0.03 m -- isn't it?
 
  • #7
dulrich said:
The block is actually rotating around in a circle with a radius of 0.03 m -- isn't it?

So are you saying I should be using 0.03 as the radius? Which formula would I use in this case?
 
  • #8
Doc Al said:
Neither of those is quite right. The first is the moment of inertia of the disk about its center. The second is almost right--Look up the parallel axis theorem.

Note that you're working on (b). What about (a)? What's the moment of inertia of the block about the axis? (Treat the block as being a small mass.)

I also tried I = .5*4.9*(.2^2) + (.2)(.17^2), but that didn't work either.
The reason I used a 2 instead of .2 before was because I figured the mass wasn't evenly distributed due to the block being added to it.
 
  • #9
pleasehelpme6 said:
I also tried I = .5*4.9*(.2^2) + (.2)(.17^2), but that didn't work either.
The reason I used a 2 instead of .2 before was because I figured the mass wasn't evenly distributed due to the block being added to it.
Forget about the block for a moment until you've finished figuring out the moment of inertia of the disk. When you look up the parallel axis theorem, you'll see why I highlighted your use of 2 (which is the mass of the block) in your earlier post.

Figure out the moment of inertia of the disk. (Which is part (b).)

Figure out the moment of inertia of the block. (Which is part (a).)

You'll need to add them in the later parts of the problem, but keep them separate for now.
 
  • #10
I understand part a, because it's just a circle with radius 0.03 m.

So Inertia of block = 2*(0.03)^2

But for the disk with the block on it, I'm confused,
From the ||Axis Theorum, I'm getting the following:

Intertia of disk = (1/2)*4.9*(.2^2) + .2*(1.7)^2
Since there's a block on it, would I do something like...

.5*4.9*(.2^2) + .2*1.7^2 + 2*(0.03)^2?
 
  • #11
pleasehelpme6 said:
I understand part a, because it's just a circle with radius 0.03 m.

So Inertia of block = 2*(0.03)^2
Good! So now you can forget about the block--you've done part (a).

But for the disk with the block on it, I'm confused,
Forget about the block--you already found its moment of inertia. Worry about the disk!
From the ||Axis Theorum, I'm getting the following:

Intertia of disk = (1/2)*4.9*(.2^2) + .2*(1.7)^2
Where did the .2 come from? The 1.7? What does the parallel axis theorem actually say?
Since there's a block on it, would I do something like...

.5*4.9*(.2^2) + .2*1.7^2 + 2*(0.03)^2?
Forget the block! :smile:

(Overall hint: The total moment of inertia of a disk with a block on it is the sum of the moment of inertia of the block plus the moment of inertia of the disk. That's why they are asking for each separately.)
 
  • #12
Doc Al said:
Good! So now you can forget about the block--you've done part (a).
Where did the .2 come from? The 1.7? What does the parallel axis theorem actually say?

I of || Axis = I of cm + Md^2

The .2 is the mass of the disk, (assuming that's what M is). Just realized I made a mistake with that one. Should be 4.9.
The .17 is the distance, (assuming that's what d is).

So with the corrected mass,
.5*4.9*(.2^2) + 4.9*1.7^2?
 
  • #13
pleasehelpme6 said:
I of || Axis = I of cm + Md^2

The .2 is the mass of the disk, (assuming that's what M is). Just realized I made a mistake with that one. Should be 4.9.
The .17 is the distance, (assuming that's what d is).

So with the corrected mass,
.5*4.9*(.2^2) + 4.9*1.7^2?
Now you've got it. (But correct the distance: 0.17, not 1.7)
 
  • #14
Oh, right...
I guess I've been staring at a computer screen for too long...

So I added the two (using the correct distances and masses) and multiplied the sum of Inertia's by (1/2)(4.2^2) for part c and got the right answer.

for part d, do i just sum the angular acceleration and the centripetal acceleration? or am i over-complicating it again?
 
  • #15
pleasehelpme6 said:
for part d, do i just sum the angular acceleration and the centripetal acceleration?
Almost. Take the vector sum of the tangential acceleration and the centripetal acceleration.
 
  • #16
right, that's the idea.

so aT = alpha*R
aC = R*w^2

and since their directions are perpendicular, sqrt(aT^2 + aC^2), right?
 
  • #17
pleasehelpme6 said:
right, that's the idea.

so aT = alpha*R
aC = R*w^2

and since their directions are perpendicular, sqrt(aT^2 + aC^2), right?
Sounds good to me!
 

1. What is the definition of inertia?

Inertia is the resistance of an object to change its state of motion or rest. It is a property of matter that is related to its mass.

2. How does the inertia of a disk and block differ?

The inertia of a disk is dependent on its mass and distribution of mass, while the inertia of a block is solely dependent on its mass. This means that a disk and a block with the same mass may have different inertias if their mass is distributed differently.

3. How does the inertia of a disk and block affect their motion?

The greater the inertia of an object, the harder it is to change its state of motion. This means that objects with larger inertias, such as disks, require more force to accelerate or decelerate compared to objects with smaller inertias, such as blocks.

4. Can the inertia of a disk and block be changed?

Yes, the inertia of an object can be changed by altering its mass or the distribution of its mass. For example, adding weight to a disk or changing its shape can affect its inertia.

5. How is the concept of inertia of disk and block used in everyday life?

The concept of inertia of disk and block is used in many everyday activities, such as driving a car or playing sports. In driving, the inertia of a car helps it maintain its speed and direction, and in sports, understanding the inertia of objects can help players make more accurate movements and shots.

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