Determine the rate of change 𝐇̇G of the angular momentum 𝐇G...

Key Concepts

Moment of Inertia

The moment of inertia is a scalar quantity that characterizes how mass is distributed with respect to a chosen axis of rotation. It quantifies an object's resistance to angular acceleration and plays a central role in determining the angular momentum and the dynamics of rotating bodies.

Torque and Newton's Second Law for Rotation

Torque is defined as the rotational analog of force and represents the effect of a force causing an object to rotate. Newton's second law for rotation states that the net external torque acting on a body is equal to the time rate of change of its angular momentum. This principle forms the basis for analyzing the rotational motion of rigid bodies.

Angular Momentum

Angular momentum is a measure of the extent of rotation of an object about a specified point or axis, combining both the distribution of mass (through the moment of inertia) and its angular velocity. It is a vector quantity and is conserved in isolated systems, making it a fundamental concept in rotational dynamics.

Transport Theorem in Rotating Reference Frames

The transport theorem provides a method for computing the time derivative of a vector quantity when observed from a rotating frame. It accounts for both the actual change in the vector and the additional effects due to the rotation of the coordinate system, making it essential for correctly determining the rate of change of angular momentum in systems with rotating components.

Transcript

00:01 Hello friends here it is given a thin rectangular plate weighing.

00:06 So weight of the plate is given 15 pounds.

00:10 Rotate about its vertical diagonal ab with angular velocity omega.

00:20 So angular velocity about its vertical diagonal av is omega.

00:57 Knowing that x -axis is perpendicular to x xx is perpendicular to plane and omega is constant this value is given constant z axis is perpendicular to plane and its value is given five radian per second we have to calculate rate of change of angular momentum that we have to calculate you can find the length of that diagonal a b h equal to root of square plus 12 square inch so it will be 15 inch let me draw the geometrical diagram of it this is y -axis this is the x -axis this is y -dhae x -axis this is y -dhax this is x -dx this is given omega this is g so resolving omega along the principal axis x -d -by -d -dase resolving angular velocity omega along the principal axis x -dose -y -dess and z omega x -des will be 12 by 15 omega that is 0 .8 into 5 radian per second or 4 radian per second and omega -y -dice 9 by 15 omega that is 9 by 15 into 5 radian per second that is 3 radian per second and omega z is 0 now moment of inertia i x test 1 by 12 15 pound so it has to be converted into mass square into 9 by 12 so it will be 021836 slug fit square.

06:55 Similarly, along principal axis by dash 1 by 12 15 upon 32 .2 whole square, 12.

07:13 So it would be 0388 to 020.

07:23 Slug into fit square moment of inertia sorry angular momentum about x -des it will be i x -dess into omega x -dess substituting the value 0 .021836 into omega -36 into omega x -d s select fit square per second similarly angular momentum of the plate about principal axis wide as so i y -dose into omega -wide s it is 0 .038 820 into 3 radian per second so it would be 0 .0 sorry 0 .0 .1 11646 select fit square per second and about the perpendicular axis z it will be iz omega z or i z is zero sorry i z it is zero so h g will be angular momentum about the center of gravity is 0 .087345 i cap dash plus 0 .11646 j -dose...

Please give Ace some feedback