A thin homogeneous square plate of mass m and side a is...

A thin homogeneous square plate of mass $m$ and side $a$ is welded to a vertical shaft $A B$ with which it forms an angle of $45^{\circ} .$ Knowing that the shaft rotates with a constant angular velocity $\omega,$ determine the angular momentum $\mathbf{H}_{A}$ of the plate about point $A$.

Key Concepts

Parallel Axis Theorem

The parallel axis theorem is a principle used to determine the moment of inertia of a body about an axis that does not pass through the center of mass. By knowing the moment of inertia about a parallel axis through the center of mass and the distance between the axes, one can compute the moment of inertia about any parallel axis, which is key in problems where rotation occurs about an off-center axis.

Inertia Tensor

The inertia tensor is a matrix that represents how mass is distributed in a rigid body relative to various axes of rotation. It allows for a complete description of the body's resistance to rotational acceleration in three dimensions and is essential for accurately computing the angular momentum when the body rotates about an arbitrary axis.

Moment of Inertia

The moment of inertia is a scalar or tensor quantity that measures the resistance of a body to changes in its rotational motion about an axis, depending on how its mass is distributed relative to that axis. It is crucial in determining the angular momentum and rotational kinetic energy of rigid bodies and is often computed through integration or using standard formulas for common shapes.

Rigid Body Dynamics

Rigid body dynamics is the branch of mechanics that deals with the motion of bodies in which deformation can be neglected. It involves analyzing both translational and rotational motion, and it relies heavily on concepts such as angular momentum, moment of inertia, and torque to predict how bodies behave under various forces and constraints.

Angular Momentum

Angular momentum is a fundamental quantity in rotational dynamics that quantifies the extent of a body’s rotational motion about a given point or axis. It is computed as the product of the body’s inertia properties and its angular velocity vector, and it plays a key role in the analysis and conservation of rotational motions in mechanical systems.

Transcript

00:01 Hello friends it is given in the problem a thin homogeneous square plate of mass am and side a is belted to a vertical soft av with its form the angle 45 knowing that soft rotates with angular velocity so angular velocity of the soft is given omega j and angular acceleration is alpha j cap we have to calculate rate of change of of the assembly about the point a let us see here we will draw its geometrical figure use principal axis by this jet jet as shown in the figure so we are drawing the pose this is the y axis already you have seen this is the dead so this is the omega in this direction and this axis we are considering the y -dice and perpendicular to this this is the z -dess and this is z and this angle is 45 so using the principal axis x -d -s as shown in the figure omega y -d -d -s will be omega cos of 45 and omega z -d s will be omega -sign of 45 and omega z -s vega x -des which is not shown in the figure is 0.

02:34 Moment of inertia of the plate about x -des is 1x3 m a square.

02:43 I -y -d -dase is 1 by 12 m a square and i -z -des will be i -x -des plus i -y -des -dess.

02:57 So 1 by 3 m a square plus 1 by 12 m a square plus 1 by 12.

03:03 M a square so it would be 5 by 12 m a square now angular momentum about a you may write i x x -dice omega x -dice i -d -y -dus omega -y -dus omega -y -dus j -d -dus i j -d -dis omega -z k -d -d -dus this is zero i -y -dddddus we have measured m a square by 12 or omega y -ddus omega cosh 45 j -dus similarly i -z we have measured 5 by 12 m a square omega sine 45 kdance now momentum of inertia along angular momentum of the system j dash has to be resolved into components so it will be cost 45 j cap minus sign 45 k cap plus 5 by 12 m a square sign and k -desk has to be resolved...

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