Do Rotational Inertia Calculations Require Integration?

In summary, the first question asked about the need for calculus in finding the rotational inertia of three identical balls fastened to a rod. The answer is yes, using the formula I=ML^2/3. The second question asked about the relationship between rotational and translational velocity in a wheel turning at a constant speed. The correct answer is that each point on the rim moves with a constant translational velocity, due to the equation v=wr and the constant radius of the wheel.
  • #1
blackout85
28
1
First questionThree identical balls, with masses M, 2M, and 3M are fastened to a massless rod of length L as shown. The rotational inertia about the left end of the rod is: Thats the layout below. Would calculus be needed in this problem (intergration) because then I am in trouble. I know the rotation at the end of rod is I=ML^2/3. Could I use that formula.

3M-----L/2----2M----L/2-----M

work:
I came up with an answer of 3ML^2/2
does that look right. I added the two end mass and lengths using the equation I=mr^2--> simply plugging in the values and adding. Would that be correct.

second question:
If a wheel turns with a costant rotational speed then: each point on its rim moves with constant roational velocity, each point on its rim moves with constant translational acceleration, the wheel turns with constant translation acceleration, the wheel turns through equal angles in equal times, the angle through which the wheel turns in each second increases as time goes on, the angle through which the wheel turns in each second decreases as time goes on.

work
I thought if the wheel turns with a constant translational velocity along the rim because of the the equation v=wr. Am I right to think since along the rim will have the same radius as in a wheel.

I would appreciate any feedback on both questions. Thank you :confused:
 
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  • #2
The answer to your first question is correct. With regards to the second question, is that copied directly from your text or have you paraphrased it?
 
  • #3
physics

I rechecked it
If a wheel turns with a constant rotational speed:
each point on its rim moves with a constant translational velocity
each point on its rim moves with a constant translational acceleration
the wheel turns through equal angles in equal times
the angle through which the wheel turns in each second increases as times goes on
the angle through which the wheel turns in each second decreases as time goes on

work:
I thought if the wheel turns with a constant translational velocity along the rim because of the the equation v=wr. Am I right to think since along the rim will have the same radius as in a wheel. This equation connects rotational velocity to translational velocity. Since the radius is the same for the edge in a wheel I thought that the answer is constant translational velocity.
 
  • #4
You are indeed correct. Although there is one further choice which is also correct...
 

Related to Do Rotational Inertia Calculations Require Integration?

1. What is rotational inertia?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to rotational motion. It is similar to mass in linear motion, but instead applies to rotational motion.

2. How do you calculate rotational inertia?

The formula for rotational inertia depends on the shape and distribution of mass of the object. For a point mass, it is calculated as I = mr^2, where m is the mass and r is the distance from the axis of rotation. For more complex objects, the formula is I = ∫r^2 dm, where the integral is taken over the entire mass distribution.

3. What is the unit of rotational inertia?

The unit of rotational inertia is kilogram-meter squared (kg·m^2) in the SI system. In the English system, it is given in units of slug-foot squared (slug·ft^2).

4. How does rotational inertia affect an object's motion?

Rotational inertia affects an object's motion by determining how easily it can be rotated. Objects with large rotational inertia require more torque to rotate, and thus have a slower angular acceleration. Objects with smaller rotational inertia can be rotated more easily and have a faster angular acceleration.

5. Can rotational inertia be changed?

Yes, rotational inertia can be changed by altering the mass or distribution of mass of an object. For example, a figure skater can increase their rotational inertia by bringing their arms closer to their body, and decrease it by extending their arms outward. This is due to the conservation of angular momentum, which states that the total momentum of a system remains constant unless acted upon by an external torque.

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