Moment of inertia of a rod at an angle to the axis

In summary, the moment of inertia of a uniform rod of mass M and length L at an angle B to the x-axis, with one end touching the axis, can be calculated by integrating along the x-axis with infinitesimal mass elements of dm = D*(dx/cosB), resulting in a moment of inertia of (1/3)*m*(L*sinB)^2. This is different from the usual solution of integrating along the rod itself, but the difference lies in the infinitesimal mass element used.
  • #1
mcheung4
22
0
suppose a uniform rod of mass M and length L is at an angle B to the x axis, one end of the the rod touching the axis. wish to find moment of inertia about x axis.

let the rod touches the axis at x=0. let D=density=M/L, and I will integrate along x axis, that means that at a distance x from the the origin, the little mass dM = D*dx is at a distance x*tanB away from the axis. then the integral I get is I = D*(x^2)*[(tanB)^2] integrated from x=0 to x=L*cosB. the answer I got is (1/3)*m*(L^2)*cosB*(sinB)^2.

I know this is different from the usual solution where one should integrate along the rod itself,but I don't understand which part of this argument went wrong?

Thanks!
 
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  • #2
If you integrate dM across the same range that you're using to calculate the moment of inertia, what value do you expect?
 
  • #3
I know this reply is quite late but I saw the problem and wanted to answer in case anyone else was confused about this.

Integrating along the x-axis instead of the rod is, of course, absolutely fine, however in this case your infinitesimal mass element dm should be dm = D*(dx/cosB), as each infinitesimal distance element dL along the length of the rod has a projection dx/cosB onto the x-axis.

With this in mind, the unwanted cosB factor of your result disappears, leaving the correct result of (1/3)*m*(L*sinB)^2
 

1. What is moment of inertia?

Moment of inertia, also known as rotational inertia, is a physical property of a rigid body that determines its resistance to rotational motion around an axis. It is analogous to mass in linear motion.

2. How is moment of inertia calculated?

The moment of inertia of a rod at an angle to the axis can be calculated using the formula I = ML^2 sin^2θ, where M is the mass of the rod, L is the length of the rod, and θ is the angle between the rod and the axis of rotation.

3. What factors affect the moment of inertia of a rod at an angle to the axis?

The moment of inertia of a rod at an angle to the axis is affected by its mass, length, and the angle at which it is held relative to the axis. It is also affected by the distribution of mass along the rod.

4. How does the moment of inertia affect rotational motion?

The moment of inertia plays a crucial role in rotational motion, as it determines how much torque is needed to rotate an object. Objects with a larger moment of inertia will require more torque to rotate, making them harder to accelerate.

5. Why is the moment of inertia important in engineering and physics?

The moment of inertia is an important concept in engineering and physics because it helps us understand the behavior of rotating objects. It is used in various applications such as designing machines, calculating the stability of structures, and analyzing the motion of celestial bodies.

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