Two Problems on Rotation of a Rigid Body

In summary, the first conversation discusses how to calculate the moment of inertia of a long rod with an LxL square cross section about its center axis. The second conversation deals with determining how far a hollow sphere will roll up a 30 degree incline after starting with a translational velocity of 5.0 m/s on a horizontal floor. The solution to this problem involves using conservation of energy and calculating both translational and rotational kinetic energy. The correct formula for rotational kinetic energy is E_{K,R}=\frac{1}{2}I \omega^2, where I is the moment of inertia and \omega is the angular velocity.
  • #1
the first

Homework Statement



Calculate the moment of inertia about a center axis of a long rod with an LxL square cross section

Homework Equations



moment of inertia (I) = the integral of the radius^2 dm

The Attempt at a Solution


I'm not really sure where to start with this one, but the I of a thin rod about its center is equal to 1/12 ML^2. This however, has a cross-section. How do I take this into account and solve the problem?
and the second:

Homework Statement



A Hollow sphere is rolling along a horizontal floor at 5.0 m/s when it comes to a 30 degree incline. How far up the incline does it roll before coming back down?

omega (angular velocity) before incline=5.0 m/s
omega at top point on incline before rolling back=0m/s


Homework Equations



Moment of Inertia of a hollow sphere= 2/3MR^2

The Attempt at a Solution


once again, I'm not sure where to begin or how to set this one up, but I think if I could figure this out I could solve the problem.
 
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  • #2
5.0 m/s cannot be angular velocity, look at the units. It is the translatoral velocity of the sphere's center of mass, I assume.
 
  • #3
that was a mistake on my part. I meant to put velocity, not angular velocity.
 
  • #4
mst3kjunkie said:
that was a mistake on my part. I meant to put velocity, not angular velocity.

Ok, now all you have to do is use energy conservation. Note only that the sphere has translatoral and rotational kinetic energy.
 
  • #5
I'm a bit unsure as to how to set up the formula, still.
 
  • #6
mst3kjunkie said:
I'm a bit unsure as to how to set up the formula, still.

Do you know how to use conservation of energy? The kinetic energy (translatoral + rotational) of the sphere must equal the potential energy of the sphere at the highest point on the incline (since the sphere's kinetic energy equals zero at that point).
 
  • #7
I've used Conservation of Energy before, but I haven't encountered the translatoral (at least the term) or rotational energy formulas yet.
 
  • #8
mst3kjunkie said:
I've used Conservation of Energy before, but I haven't encountered the translatoral (at least the term) or rotational energy formulas yet.

[tex]E_{K,T}=\frac{1}{2}mv^2[/tex], and
[tex]E_{K,R}=\frac{1}{2}I \omega^2[/tex].
 
  • #9
okay, now I"ve gotten it to

(25/2)m+(1/3)mr^2(omega)^2 = 9.8mh

am I on the right track?
 
  • #10
Yes, but note only that the moment of inertia of the sphere must be calculated with respect to the bottom point, since the translational velocity at that point equals zero, so it is the center of rotation. This link may be a useufl reference: http://www.physics.upenn.edu/courses/gladney/mathphys/java/sect4/subsubsection4_1_4_3.html" [Broken]. I hope you'll get everything right now.
 
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1. What is the definition of a rigid body?

A rigid body is an object that does not change its shape or size when subjected to external forces. It can be thought of as a collection of particles that are fixed in relation to each other.

2. What is rotation of a rigid body?

Rotation of a rigid body is the movement of the body around a fixed axis or point. This movement is characterized by an angular displacement, velocity, and acceleration.

3. What is the difference between angular displacement, velocity, and acceleration?

Angular displacement is the change in the orientation of a rigid body, measured in radians or degrees. Angular velocity is the rate of change of angular displacement, while angular acceleration is the rate of change of angular velocity.

4. What are the two problems on rotation of a rigid body?

The two problems on rotation of a rigid body are the kinematics problem and the dynamics problem. The kinematics problem involves determining the angular displacement, velocity, and acceleration of a rigid body, while the dynamics problem involves calculating the forces and torques acting on the body.

5. How is the solution to these problems useful in real-world applications?

The solution to problems on rotation of a rigid body is useful in many real-world applications, such as understanding the motion of objects in space, designing mechanical systems, and analyzing the movement of athletes in sports. It is also essential in fields such as robotics, aerospace engineering, and biomechanics.

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