Calculate the moment of inertia of a straight rod of length λand having linear mass density λ=Λs

Calculate the moment of inertia of a straight rod of length...

Key Concepts

Moment of Inertia

This is a physical quantity that quantifies a body's resistance to rotational acceleration about a given axis. It is calculated by integrating the product of small mass elements and the square of their perpendicular distances from the axis of rotation, fundamentally representing how mass is distributed relative to the axis.

Variable Mass Density

When mass is distributed non-uniformly along an object, the mass density is not constant, but a function of position. This variation requires using the specific density function in integrals to correctly account for the changing amount of mass per unit length, area, or volume in calculations such as the moment of inertia.

Center of Mass

The center of mass of an object is the point where the entire mass of the system can be considered to be concentrated for the purpose of analysis of translational or rotational motion. Determining the center of mass is essential when calculating moments of inertia about axes that do not pass through a uniform section of the mass distribution.

Integration in Mechanics

Integral calculus plays a crucial role in mechanics, especially for calculating properties like the moment of inertia. It involves summing the contributions of infinitesimally small mass elements over the body, which is indispensable when dealing with continuous, nonuniform mass distributions.

Axis Orientation

The moment of inertia depends not only on the mass distribution but also on the orientation of the rotation axis relative to the object. The angle between the axis and principal directions of the body influences the effective distances that mass elements have from the axis, thereby altering the moment of inertia calculation.

Transcript

00:02 Hello everyone, we are going to understand this question here given in the question mass density linear mass density of rod it is represented by lambda which is equal to a into s let s is equal to x so we can write lambda is equal to a into x so let there is a small part d x so let the partial length of rod is d x partial length of rod is equal to d x so mass of partial length mass of partial length dm is equal to lambda into d x mass linear mass density into length of rod linear mass density is a into x into d x now calculating a center of mass of rod so let center of mass of rod is represented by x c o m which is equal to integration of dm into x upon integration of dm.

01:40 Here value of dm is a x into d x upon upon dm is equal to a x into d x.

01:59 Here we forgot to multiply by x now doing further calculation a x square x upon x x into d x x into d x now after calculation we will get x q upon 3 upon x squared upon 2 here integration limit is from 0 to l now doing further calculation we will get l q upon 3 whole upon l square upon 2 after calculation we will get 2 l 3.

02:55 This is the center of mass of rod.

03:00 Now calculating the movement of inertia of rod, d -i is equal to d -m into x square.

03:11 Not x because we have taken dm from the one end.

03:18 So here we can take r square.

03:22 Now r is equal to distance from center of mass is equal to 2l upon 3.

03:31 Minus x.

03:33 So we can write d i is equal to d m into 2l upon 3 minus x square.

03:44 But here r is along to rod but we have to find the perpendicular distance from the axis of rotation.

03:55 So we can take it a square but here a is equal to let's see the diagram.

04:16 This is the length r this length is r and we have to find the a.

04:25 Now in triangle we can see this angle is alpha then this one is also alpha.

04:35 This is r and this is a...

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