Rotational Dynamics disk of radius Question

When you drop the ring on the disk, it becomes part of the system. The motion of the disk is still simple harmonic motion, but the period is changed. You have the right idea for a) and c) in the "at the end of the swing" case. The period increases while the amplitude and energy remain the same because the extra mass of the ring increases the moment of inertia of the system. For b) you are given that the disk is at rest when the ring is added. This means the ring doesn't add any energy to the system. So the amplitude is the same. However, the period increases because the extra mass increases the moment of inertia. For the second case,
  • #1
Hoofbeat
48
0
Please could someone check my answers to the following and advise me on any of the bits I'm stuck on. Thanks

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Q. A disk of radius a and mass m is suspended at its centre by a vertical torsion wire which exerts a couple [tex]\fontsize{5} -c\theta [/tex] on the disk when it is twisted through an angle [tex]\fontsize{5} \theta [/tex] from its equilibrium position. Show that the oscillations of the disk are simple harmonic, and obtain an expression for the period [I've done this proof so that bit's fine]

A wire ring of mass m and radius a/2 is dropped onto the disk and sticks to it. Discuss what happens to (a) the period, (b) the amplitude, (c) the energy of the oscillations for the 2cases where the ring is dropped on (i) at the end of the swing when the disk is instantaenously at rest (ii) at the midpoint of the swing when the disk is moving with is maximum angular velocty. Assume the ring and disk are concentric.
====

i) At end of swing
a) The period increase as the moment of inertia has increased
b) The amplitude increases as the velocity remains same as before, but the time period has increased
c) The energy is unchanged as at the end of the swing where v=0, the object only has potential energy.

ii) At midpoint of motion
a) The period increases due to increase in moment of inertia
b) ?
c) PE=0. There is a change in energy as the extra disk increases friction so the KE is lower?

Thanks.
 
Last edited:
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  • #2
Hoofbeat said:
Please could someone check my answers to the following and advise me on any of the bits I'm stuck on. Thanks

=====
Q. A disk of radius a and mass m is suspended at its centre by a vertical torsion wire which exerts a couple [tex]\fontsize{5} -c\theta [\tex] on the disk when it is twisted through an angle [tex]\fontsize{5} \theta [\tex] from its equilibrium position. Show that the oscillations of the disk are simple harmonic, and obtain an expression for the period [I've done this proof so that bit's fine]

A wire ring of mass m and radius a/2 is dropped onto the disk and sticks to it. Discuss what happens to (a) the period, (b) the amplitude, (c) the energy of the oscillations for the 2cases where the ring is dropped on (i) at the end of the swing when the disk is instantaenously at rest (ii) at the midpoint of the swing when the disk is moving with is maximum angular velocty. Assume the ring and disk are concentric.
====

i) At end of swing
a) The period increase as the moment of inertia has increased
b) The amplitude increases as the velocity remains same as before, but the time period has increased
c) The energy is unchanged as at the end of the swing where v=0, the object only has potential energy.

ii) At midpoint of motion
a) The period increases due to increase in moment of inertia
b) ?
c) PE=0. There is a change in energy as the extra disk increases friction so the KE is lower?

Thanks.

You need to look at 1b again. The disk is at rest with the torsion wire twisted when the ring gets added. Does that make the wire twist more?

iib. What will happen to the angular velicocity at impact. This is the position of maximum velocity for the subsequent motion.

iic follows form iib. Whe you get b, think about c again
 
  • #3
OlderDan said:
You need to look at 1b again. The disk is at rest with the torsion wire twisted when the ring gets added. Does that make the wire twist more?

iib. What will happen to the angular velicocity at impact. This is the position of maximum velocity for the subsequent motion.

iic follows form iib. Whe you get b, think about c again

So is 1b) The moment of inertia increases and as no external torques are acting, angular momentum must be conserved. We know [tex]L=I\omega[/tex] which is the same as [tex]L=IV/r[/tex] so an increase in the inertia will cause the velocity to decrease and thus the amplitude will remain the same (as slower speed and longer time period)?
 
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  • #4
Angular momentum is not conserved in this problem. This problem is analogous to a mass on a spring, where the spring bobs up and down, or to a pendulum. In fact this system is called a torsion pendulum. The torsion wire produces a torque in proportion to the angle through which it is twisted. The angular displacement of the disk goes back and forth. There is an anergy associate with the torsion, and there is a back and forth between stored energy and kinetic energy. The equations for the motion of this system are very similar to the equations for a mass on a spring.
 

1. What is rotational dynamics and how does it apply to a disk of radius?

Rotational dynamics is the study of the motion of rotating objects, such as a disk. It involves understanding how forces and torques act on a rotating object and how they affect its motion. For a disk of radius, rotational dynamics can be applied to calculate its angular acceleration, angular velocity, and rotational energy.

2. What is the relationship between torque and angular acceleration for a disk of radius?

The relationship between torque and angular acceleration for a disk of radius is given by the equation τ = Iα, where τ is the torque, I is the moment of inertia of the disk, and α is the angular acceleration. This means that the larger the moment of inertia, the smaller the angular acceleration will be for a given torque.

3. How does the distribution of mass affect the rotational dynamics of a disk of radius?

The distribution of mass affects the rotational dynamics of a disk of radius by influencing its moment of inertia. The moment of inertia is a measure of an object's resistance to rotational motion, and it depends on the mass and its distribution relative to the axis of rotation. A disk with a larger moment of inertia will require more torque to achieve the same angular acceleration as a disk with a smaller moment of inertia.

4. Can you explain the concept of angular momentum in the context of a disk of radius?

Angular momentum is a measure of an object's rotational motion and is defined as the product of its moment of inertia and angular velocity. In the context of a disk of radius, angular momentum is conserved, meaning it remains constant as long as there are no external torques acting on the disk. This means that if the moment of inertia decreases, the angular velocity must increase to maintain the same angular momentum, and vice versa.

5. How can the equations of rotational dynamics be applied to solve real-world problems involving a disk of radius?

The equations of rotational dynamics can be applied to solve real-world problems involving a disk of radius by using the principles of conservation of angular momentum and energy. By considering the forces and torques acting on the disk, along with its moment of inertia and initial conditions, we can use these equations to determine its final state of motion. This can be useful in designing machines, understanding the motion of celestial bodies, and many other applications.

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