A small disk and a bigger half disk

In summary, "A small disk and a bigger half disk" explores the geometric relationship between a small circular disk and a larger semi-circular disk. It delves into their dimensions, properties, and potential applications in various fields such as mathematics and engineering, highlighting the significance of understanding these shapes in practical scenarios.
  • #1
billtodd
137
33
Homework Statement
In the pic attached please look at the last figure below.
We have a half disk with moment of inertia ##I_B##, and as the half disk completes a complete cycle of 180 degrees or ##\pi## if you prefer.
A small disk with mass ##m## which is smaller than the mass of the half disk, and velocity ##v_0##, it's given that the mass of the small disk is much smaller than that of the half disk.
What will be the angular speed of the half disk
Relevant Equations
Torque here.

I believe the equation of torques is: ##mgR\sin\theta- MR\dot{\theta}=I_B \ddot{\theta}##
And that of conservation of linear momentum is: ##mv_0=I_B\dot{\theta}+mv_1##
I need to find ##v_1## and I know what are the initial conditions: ##\theta(0)=\pi## and ##\dot{\theta}(0)=0##.

Then what is ##v_1## and how to find it?

Thanks!
צילום מסך 2024-03-24 ב-12.02.16 (1).png
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
You have not described adequately what happens here. Is this a collision? If it is and if the collision is along the axis of symmetry of the half-disk (as appears to be the case), what reason does the half disk have to rotate clockwise as opposed to counterclockwise? The situation would be different if the collision were to take place off-axis. Do you see why and how?
 
  • #3
I think it should be ##-\pi##, because the motion of the half big disk is counterclockwise.
 
  • #4
billtodd said:
I think it should be ##-\pi##, because the motion of the half big disk is counterclockwise.
What quantity should be ##-\pi##? That's just a number without units. What reason do you have to say that the motion of the half disk is counterclockwise as opposed to clockwise? What criterion do you use to decide if it's one way or the other?
 
  • #5
@kuruman
It can't go up beyond the horizontal line because of gravity, its motion would be chaotic and not a simple harmonic. Or so I think.
 
  • #6
billtodd said:
@kuruman
It can't go up beyond the horizontal line because of gravity, its motion would be chaotic and not a simple harmonic. Or so I think.
That's more a riddle than a problem statement!
 
  • Like
Likes berkeman
  • #7
I still don't understand what the physical situation or the statement of the problem are. Can you translate the text that's in the figure? According to our rules, postings must be in English.
 
  • #8
kuruman said:
I still don't understand what the physical situation or the statement of the problem are. Can you translate the text that's in the figure? According to our rules, postings must be in English.
I'll do the translation later on tomorrow. I got to regain my physics problem solving skills, they got really rusty.

It's not about that I don't remember the equations, I just never quite know how to choose the coordinate axes, etc.
 
  • #9
billtodd said:
It's not about that I don't remember the equations, I just never quite know how to choose the coordinate axes, etc.
We are here to help you remember how to do it but we have to know the details of the task.
 
  • Like
Likes billtodd
  • #10
Does the notation ##I_B## mean the half disk is hinged at B? If so, how is it moving at the start?
 

FAQ: A small disk and a bigger half disk

What are the primary differences in the physical properties between a small disk and a bigger half disk?

The primary differences lie in their dimensions, mass, and moments of inertia. A small disk is fully circular and typically has a smaller radius and mass compared to a bigger half disk, which is a semicircular object with a larger radius and potentially greater mass. The distribution of mass also differs, affecting their rotational dynamics.

How do the moments of inertia compare between a small disk and a bigger half disk?

The moment of inertia for a small disk about its central axis is given by \( I = \frac{1}{2} m r^2 \), where \( m \) is the mass and \( r \) is the radius. For a bigger half disk about an axis through its flat edge, the moment of inertia is \( I = \frac{1}{4} m R^2 + \frac{1}{3} m R^2 = \frac{7}{12} m R^2 \), where \( m \) is the mass and \( R \) is the radius. The bigger half disk generally has a larger moment of inertia due to its greater mass and radius.

How does the center of mass differ between a small disk and a bigger half disk?

The center of mass for a small disk is at its geometric center. For a bigger half disk, the center of mass is located along the central axis but shifted towards the curved edge. Specifically, it is at a distance of \( \frac{4R}{3\pi} \) from the flat edge, where \( R \) is the radius of the half disk.

In what ways do the shapes of a small disk and a bigger half disk affect their stability and equilibrium?

A small disk, being symmetrical, has uniform stability and can balance easily on its edge or flat surface. A bigger half disk, however, has an asymmetrical shape that affects its equilibrium. It tends to be more stable when resting on its flat edge but less stable if balanced on the curved edge due to the uneven distribution of mass.

What are the practical applications of small disks and bigger half disks in engineering and design?

Small disks are commonly used in applications requiring rotational symmetry, such as wheels, gears, and flywheels. Bigger half disks, with their unique shape, are used in applications where partial rotational symmetry is beneficial, such as in certain types of cams, architectural elements, and specific mechanical components where space constraints or specific load distributions are required.

Similar threads

Perfectly inelastic collision between Projectile and a hinged disk
Replies
26
Views
682
Question about the current components inside the Corbino disk
Replies
11
Views
636
Stretched spring attached to the center of a pure rolling disk
Replies
11
Views
556
What is the final angular displacement of a disk hit by two masses?
Replies
3
Views
1K
A rod that falls while rotating from the end of a table
Replies
11
Views
599
Potential and Electric Field near a Charged CD Disk
Replies
2
Views
1K
Electromagnetic Induction of a Disk
Replies
5
Views
2K
Equation of Motion for a Disk inside a Rotating Ring
Replies
5
Views
3K
What minimum speed do we need to give a charged ball?
Replies
12
Views
875
Electric Field due a charged disk
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top