Rotational Inertia with varying axis of rotation?

In summary, the rotational inertia of a rigid body with four identical particles of mass 0.475 kg each placed at the vertices of a 1.73 m x 1.73 m square and held by four massless rods is 0.948 kg*m^2 when the rotational axis passes through the midpoints of opposite sides and lies in the plane of the square. To calculate the rotational inertia for the other two locations of the rotational axis, the moment of inertia equation I = (1/12)M(a^2 + b^2) can be used with the appropriate values for M, a, and b. Further assistance may be needed to determine the specific values for each location.
  • #1
Javasauce
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Homework Statement



Four identical particles of mass 0.475 kg each are placed at the vertices of a 1.73 m x 1.73 m square and held there by four massless rods, which form the sides of the square. What is the rotational inertia of this rigid body about an axis that (a) passes through the midpoints of opposite sides and lies in the plane of the square, (b) passes through the midpoint of one of the sides and is perpendicular to the plane of the square, and (c) lies in the plane of the square and passes through two diagonally opposite particles?

Homework Equations


Icom = (1/12)M(a^2 + b^2)

The Attempt at a Solution


M = 0.475 * 4 = 1.9kg
I = (1/12)M(a^2 + b^2) = (1/12)(1.9)(1.73^2 * 2) = 0.948 kg*m^2

I believe this answer would be correct for part (a), but how do I go about calculating the rotational inertia for the other two locations of the rotational axis? Any help would be greatly appreciated, thanks!
 
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  • #2
bump, can anyone lend any advice to this question? :-\
 

Related to Rotational Inertia with varying axis of rotation?

1. What is rotational inertia with varying axis of rotation?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to change in its rotational motion. It is affected by the mass and distribution of the object's mass around its axis of rotation. When the axis of rotation changes, the rotational inertia also changes.

2. How is rotational inertia with varying axis of rotation calculated?

The formula for rotational inertia is I = ∫r^2 dm, where I is the rotational inertia, r is the distance from the axis of rotation to the mass element, and dm is the mass element. This integral is taken over the entire object to calculate the total rotational inertia. When the axis of rotation changes, the distance from the axis to the mass element changes, resulting in a different rotational inertia value.

3. What is the relationship between rotational inertia and axis of rotation?

The relationship between rotational inertia and axis of rotation is inverse. When the axis of rotation is closer to the mass, the rotational inertia decreases. When the axis of rotation is further from the mass, the rotational inertia increases. This is because the distance from the axis of rotation affects the distribution of the object's mass and its resistance to change in rotational motion.

4. How does rotational inertia with varying axis of rotation affect an object's motion?

The rotational inertia with varying axis of rotation affects an object's motion by determining how difficult it is to change its rotational motion. Objects with a larger rotational inertia will require more force to change their rotational motion, while objects with a smaller rotational inertia will require less force. The axis of rotation also affects the object's stability and balance.

5. What are some real-life applications of understanding rotational inertia with varying axis of rotation?

Understanding rotational inertia with varying axis of rotation is important in many fields, including engineering, physics, and sports. In engineering, it is used to design and optimize the performance of rotating machinery such as motors and turbines. In physics, it helps explain the behavior of objects in rotational motion, such as planets orbiting the sun. In sports, it is essential for athletes to understand rotational inertia to perform certain movements, such as figure skaters performing spins or gymnasts performing flips on uneven bars.

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