Finding Moment of Inertia for Rotated Solid: y=sinx, x-axis, [0,pi]

In summary, to find the moment of inertia of a solid formed by rotating the curve of y=sinx about the x-axis, one can use the formula \frac{1}{2}\rho \pi y^4 dx where \rho is the volume density of the solid. To remove the \rho term and leave the answer in terms of the total mass M, one can calculate the volume of revolution V and substitute \rho = \frac{M}{V}.
  • #1
devious_
312
3
How do I find the moment of inertia of a solid formed by rotating the curve of y=sinx about the x-axis in the interval [0, pi]?

I've tried to set up integrals by summing up cylinders parallel to the y-axis but to no avail.
 
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  • #2
I'm assuming it's the moment of inertia of the solid about the x-axis that the question is asking for.

Let the volume density of the solid of revolution be [itex]\rho[/itex]. Then a cylindrical element of mass [itex]dm[/itex] is defined by [itex]\rho \pi y^2 dx[/itex]. The moment of inertia of that element about the x-axis is defined by [itex]\frac{1}{2}y^2dm = \frac{1}{2}\rho \pi y^4 dx[/itex]. Substitute [itex]y = \sin x[/itex] and integrate over the required bounds and you have the answer. To remove the [itex]\rho[/itex] term and leave your answer purely in terms of the total mass [itex]M[/itex], just calculate the volume of revolution [itex]V[/itex] the usual way and put [itex]\rho = \frac{M}{V}[/itex].
 
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  • #3
Yeah, that worked!

What I did was EXACTLY the same as your method, except I multipled dm by x^2 instead of y^2. Oops.. :shy:

Thank you. :smile:
 

What is moment of inertia?

Moment of inertia is a measure of an object's resistance to rotational motion. It is also known as angular mass or rotational inertia. It is calculated using the object's mass and its distribution around an axis of rotation.

How is moment of inertia different from mass?

Moment of inertia is different from mass because it takes into account not only the object's mass, but also its distribution around an axis of rotation. While mass is a measure of an object's resistance to linear motion, moment of inertia is a measure of its resistance to rotational motion.

What factors affect the moment of inertia of a solid?

The moment of inertia of a solid is affected by the object's mass, shape, and distribution of mass around the axis of rotation. Objects with larger mass, greater distance from the axis of rotation, and more spread out mass distribution have higher moment of inertia.

How is moment of inertia calculated for a solid object?

The moment of inertia of a solid object can be calculated by using the formula I = ∫r²dm, where r is the distance from the axis of rotation and dm is the mass element. This formula can be applied to different types of solids, such as cylinders, spheres, and rectangular prisms, by adjusting the limits of integration and the mass element.

Why is moment of inertia important in physics and engineering?

Moment of inertia is important because it helps us understand and predict the behavior of rotating objects. It is used in various applications, such as designing machines and structures that involve rotational motion, calculating the stability of vehicles, and determining the rotational energy of objects. It also plays a crucial role in explaining phenomena such as gyroscopic motion and the conservation of angular momentum.

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