The pendulum consists of a 8 -kg circular disk A, a 2-kg circular disk B, and a 4-kg slender rod

step 1 Hello and welcome to this video solution of numerate. Here it's given that there is a pendulum t

The pendulum consists of a 8 -kg circular disk A, a 2-kg...

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to angular acceleration about a given axis. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. In composite bodies, the total moment of inertia is the sum of the moments of inertia of each individual component calculated about the same axis, sometimes requiring adjustments with the parallel axis theorem if the components’ axes do not coincide with the rotation axis.

Radius of Gyration

The radius of gyration is a scalar quantity that represents the distribution of the mass of an object around an axis, such that the moment of inertia can be expressed as the product of the total mass and the square of the radius of gyration. It provides a compact way to describe the spread of mass in a rotating object, which is especially useful when comparing different shapes or composite bodies.

Composite Bodies

Composite bodies consist of multiple individual components that together form a larger structure. When analyzing such systems, one sums the individual contributions (like moments of inertia) from each component to determine properties of the entire system. This approach requires careful consideration of each component’s mass distribution and may involve applying the parallel axis theorem if the centers of mass of the components do not lie on the common rotation axis.

Parallel Axis Theorem

The parallel axis theorem is a useful tool for computing the moment of inertia of an object about an axis that is parallel to an axis through its center of mass. It states that the moment of inertia about the new axis is equal to the moment of inertia about the center of mass axis plus the product of the mass of the object and the square of the distance between the two axes. This theorem is essential when dealing with composite bodies where the components’ centers of mass are offset from the main axis of rotation.

Transcript

00:01 Hello and welcome to this video solution of numerate.

00:04 Here it's given that there is a pendulum that consists of 8 kg circular disk.

00:08 Right.

00:09 So this a let's say there is a 2 kg circular disk b or maybe let me write ma that is equal to 8 kgs.

00:20 Mass of b is equal to 2 kg whereas there is a 2 kgs cylinder rod mc i am taking.

00:30 Okay.

00:31 That is equal to 4 kgs.

00:33 Right.

00:34 Now based on this you have to determine the radius of gyration of the pendulum about an axis perpendicular to the page and passing through the point o.

00:41 Right.

00:44 Kind of like this axis you have to take.

00:46 So what you can do is you can calculate the moment of inertia of this entire system.

00:51 Right.

00:51 About the point o or the axis about o.

00:57 Right.

00:58 So what you can do is i you can start by writing the moment of inertia of this disk about the centroidal axis passing through the page.

01:07 Right.

01:08 So i1 will be equal to 1 by 2 mra square.

01:12 Right.

01:13 Now the diameter is given as 0 .4.

01:16 Right.

01:16 Which is equal to 1 by 8 mada square by da square.

01:24 Right.

01:25 Now if you plug in the values then ma is 8 sorry 1 by 8 times 8 and da is 0 .4.

01:35 Right.

01:38 Kg meter square.

01:39 Right.

01:39 Which is equal to you have got 0 .16 kg meter square is the value i1 about the center.

01:52 Right.

01:54 Now next let's calculate the moment of inertia of this rod i2 let's say i'm taking.

02:05 Okay.

02:05 Which is ml square or mcl square by 12 about the center which is 4 times of 1 .5 squared over 12.

02:18 This is equal to 0 .75 kg meter square.

02:24 Right.

02:26 Next the other disk i3 let's say which is equal to 2 times of 0 .2 squared over 8.

02:35 Same formula you have to apply which is 0 .01 kg meter square...

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