Rigid Body Rotational Motion

In summary, in this conversation, the question asks about the angular velocity of a uniform rod falling on a rough horizontal floor. The solution involves calculating angular acceleration using the equation \alpha\,=\,\dot\omega\,=\,\ddot\theta and then finding the integral of this equation to solve for angular velocity. The final equation will be a function of theta.
  • #1
moo5003
207
0
Question:
"A uniform rod of mass m and length 2a stands vertically on a rough horizontal floor and is allowed to fall. Assuming that slipping has no occured, write the angular velocity of the rod as a function of the angle Theta the rod makes with the vertical."

WORK DONE:
Ic = (ml^2)/3
Torque = mgd
d = asin(theta)
Torque = angular accel * I

mgasin(theta) = ang accel * (ml^2)/3
l = 2a

gasin(theta) = ang accel * (m(2a)^2)/3
gsin(theta) = 4/3 * ang acell * a

Ang Accel = 3/(4a) * gsin(theta)

MAIN QUESTION: How do I substitute Angular Acceleration such that I can find an equation that solves angular velocity?
 
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  • #2
Angular acceleration [itex]\alpha\,=\,\dot\omega\,=\,\ddot\theta[/itex]

Angular velocity [itex]\omega\,=\,\dot\theta[/itex]
 
  • #3
Astronuc said:
Angular acceleration [itex]\alpha\,=\,\dot\omega\,=\,\ddot\theta[/itex]
Angular velocity [itex]\omega\,=\,\dot\theta[/itex]

I considered using calc to solve for this. But the integral of Angular Acceleration is a function of time. I'm unsure how I can incorporate that together and produce an equation as a function of Theta.

W = Integral [ 3/4 * 1/a * gsin(theta) ] dt

Would I just slap on a variable T and give that as my answer (There is no initial angular velocity right)? Or can I somehow subtitute T as a function of Theta?
 
Last edited:

1. What is rigid body rotational motion?

Rigid body rotational motion is the movement of a solid object around a fixed axis, where all points on the object move in circular paths at the same time and with the same angular velocity.

2. What is the difference between rotational motion and translational motion?

Rotational motion involves movement around a fixed axis, while translational motion involves movement in a straight line. Additionally, rotational motion is described using angular quantities such as angular velocity and angular acceleration, while translational motion is described using linear quantities such as velocity and acceleration.

3. How is angular velocity calculated?

Angular velocity is calculated by dividing the change in angular displacement by the change in time. It is represented by the symbol ω and its unit is radians per second (rad/s).

4. What is the moment of inertia in rotational motion?

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass of the object, the distribution of the mass around the axis of rotation, and the distance of the mass from the axis. It is represented by the symbol I and its unit is kilogram-meter squared (kg·m²).

5. How does the rotational motion of a rigid body affect its kinetic energy?

The kinetic energy of a rigid body in rotational motion is given by the formula KE = ½Iω², where I is the moment of inertia and ω is the angular velocity. This means that the kinetic energy increases as the moment of inertia or angular velocity increases, and decreases as the rotational speed decreases.

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