A uniform square plate with side a=225 mm is hinged at points A and B to a clevis that rotates w

A uniform square plate with side a=225 mm is hinged at...

Key Concepts

Free-Body Diagram Analysis

Constructing and analyzing a free-body diagram (FBD) is a fundamental step in mechanics. It involves isolating the body of interest and depicting all external forces and moments acting on it. For a system with a rotating plate, the FBD helps to visualize gravitational forces, centrifugal forces, and reaction forces at the hinges, thereby facilitating the formulation and solution of equilibrium equations.

Geometric Relationships in Rigid Body Dynamics

This concept involves understanding how the geometry and mass distribution of a rigid body, such as a square plate, influences its center of mass and moment arms relative to the applied forces. Such geometric considerations are critical when determining how forces translate into moments about pivot points or hinges, ultimately affecting the orientation and stability of the body under various loading and motion conditions.

Rotational Kinematics

Rotational kinematics deals with the description of motion involving angles, angular velocities, and accelerations. In this context, a constant angular velocity implies uniform circular motion, which affects how inertial forces such as centrifugal force are calculated. This concept is further used to understand the relation between the rotation of the supporting structure and the orientation of the attached body.

Hinge Constraints and Reaction Forces

Hinges restrict the motion of a body by providing constraints that allow rotation only around specific axes, while also transmitting reaction forces and moments that maintain equilibrium. When a body is connected by hinged supports, the analysis needs to account for the forces these hinges exert in response to other loads, thus playing a central role in setting up the equations of motion or static equilibrium.

Centrifugal Force in Rotating Systems

When analyzing bodies in a rotating system, a fictitious force known as the centrifugal force must be considered. This inertial force acts radially outward from the axis of rotation and is proportional to the square of the angular velocity and the mass of the body. Understanding how the centrifugal force interacts with gravitational force and other applied loads is key to determining equilibrium positions or stability limits in rotating bodies.

Rigid Body Equilibrium

This concept involves ensuring that the net forces and net moments acting on a rigid body are balanced, so that the object remains in static or dynamic equilibrium. In problems involving hinges or supports, determining the conditions for equilibrium is essential to translate the physical constraints of the system into mathematical equations, which relate applied forces, moments due to gravity, and inertial (often centrifugal) forces when the body is in a rotating reference frame.

Transcript

00:01 Hello everyone as soon in the figure there is a uniform circular plate of radius 120 millimeter that is 0 .12 meter hinge at a and b to a clavis which rotate with constant angular velocity omega with vertical axis calculate first part angle beta that the plane formed with the horizontal for omega to be 15 radian per second.

00:58 Second part, omega max for the plane remains vertical.

01:23 Consider the centroidal axis x -dus -by -des -z -z -jad.

01:49 Sorry not centroidal axis.

01:52 Consider the axis see x -d -y -dase -s.

02:05 Suppose this is by -dess, this is x -des.

02:13 So this angle will be also beta.

02:18 Let us see here now.

02:20 Moment of inertia of the semicircular plate having the mass m, i x, dash, will be mr squared by 2, iy dash will be mr square by 2 and due to symmetry i x -y -dice, i -z -x -x -x -x -dase is 0.

03:14 Now angular momentum about the c at c we are finding i x does omega x dash i cap i y -dice omega y -d s d s i i x -d s i x is m r squared by 4 radius we are taking the symbol is smaller so here write smaller not capital r you may take that is your choice it is immaterialistic mr square by four radius we are taking the symbol by 4 omega sine of beta i -d s mr -square by 2 omega cos of veta z so we can write 1 by 4 mr square omega common then we can write minus sine of veta i -d s plus 2 cos of beta z say this is equation 1 angular velocity of reference frame c by dash z is omega which is minus omega sine of beta i -dus plus omega cos of beta z z so rate of change of angular momentum will be change of angular momentum rate of change of angular momentum in the reference frame c x -y -dase z plus omega cross angular momentum about c this term is 0 minus omega sine of veta i -d s plus omega cos of veta j -d s cross mr square by 4 into to omega minus sine of veta i -d -s plus cos of veta z -s.

08:06 Now simplify it, so it could be written as minus 1 by 4 m r square omega square sine of beta, cos of beta, k -caf...

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