The mass o⌈given Γ-shaped uniform wirc framc is m . Calculatc the moment of inerlia of the wire

The mass o⌈given Γ-shaped uniform wirc framc is m ....

Key Concepts

Integration for Continuous Bodies

For continuous bodies, the moment of inertia is determined by integrating over the body’s mass distribution. This involves expressing small mass elements in terms of the density and differential length or volume, and then integrating these contributions weighted by the square of their distances from the axis of rotation.

Composite Body Analysis

When an object is made up of several distinct segments or shapes, the overall moment of inertia can be found by calculating the moment of inertia for each segment and then summing them. In cases where the axis of rotation does not pass through the center of mass for each segment, the parallel axis theorem may be used to adjust the calculation for each part before summing.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion around a particular axis. It quantifies how mass is distributed with respect to that axis and is calculated by summing or integrating the products of each mass element with the square of its distance from the axis of rotation.

Uniform Mass Distribution

Uniform mass distribution means that the mass per unit length (in the case of wires or rods) or volume (in three-dimensional objects) is constant throughout the object. This simplifies the calculation of the moment of inertia, as the density can be taken as a constant factor during integration over the object.

Transcript

00:01 Hello everyone, we are going to understand this question here given in the question given mass of e -shaped wire mass is equal to m and total length of the wire from the diagram we can see total length of the wire total length of e -shaped wire that is equal to length of ac plus length of of plus length of ob plus length of bd now 2l plus l plus l plus l plus l plus l.

01:18 Now adding all these terms we will get 7 l so mass density of e -shaped wire mass density of e -shaped wire, that is total mass upon total length, that is 7l.

01:53 This is the mass density of e -shaped wire.

01:59 So, mass of ac, that is represented by m -ac, which is equal to m -7l into 2l, that is equal to 2m -7.

02:15 Mass of a -o which is equal to m upon 7l into l.

02:28 So after calculation we will get m upon 7.

02:33 Mass of of that is equal to m upon 7l into l.

02:42 After calculation we will get m upon 7 mass of ob that is m upon 7 into l so after calculation we will get m upon seven similarly mass of bd that is m upon seven two l so after calculation we will get two m upon seven now calculating the movement of inertia we know that moment of inertia of wire or or or round m i is equal to sorry moment of energy of oaf wire or oaf rod that is 1 upon 3 m by 7 into l square after calculation we will get m l square upon 21 similarly moment of inertia of oa that is also 1 upon 3 into a m by 7 into l square so after calculation we will get m l square upon 21 moment of inertia of wire ob that is 1 upon 3 into m upon 7 into l square so after calculation we will get m l square upon 21 similarly movement of inertia of wire ac iac now here applying the parallel axis theorem so movement of inertia of ac is equal to moment of inertia of center of mass plus m d square moment of inertia of center of mass of rod is equal to 1 upon 2l into m 2m upon 7 into 2l 2l square plus distance between center of mass and o f this is the length 2l and this is length l and this is l and this is 2 l so center of mass of rod will be at this place this length is l and this is l and this is l so this length o e -d -dice is equal to square root of l and square plus l square a dash o is equal to l square plus l square so that we will get l square root 2 so here d is equal to square root 2 so here 2m upon 7 into l square root 2 square so after calculation we will get 8ml square upon 84 plus 4 ml square upon 7.

07:01 Now taking the lcm, so here 12 into 7 will be the lcm 12 into 7 and 8m plus 48ml square.

07:28 So after doing further calculation we will get 56ml square upon 12 into 7.

07:45 So after calculation we will get 8m .m.

07:51 Square upon 12...

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