. If the gap between C and the rigid wall at D is initially 0.15 mm , determine the support reac

. If the gap between C and the rigid wall at D is initially...

Key Concepts

Material Properties and Constitutive Laws

Material properties, such as stiffness and yield strength, define how a material responds to applied loads. Constitutive laws, like Hooke's law for linear elastic materials, describe the relationship between stress and strain in these materials. These concepts are crucial in predicting the deformation behavior of structural elements under load and ultimately affect the determination of reaction forces in the assembly.

Contact Constraints

Contact constraints govern the interaction between surfaces that come into contact, ensuring that no interpenetration occurs. In problems where an initial gap exists that eventually closes due to loading, contact constraints must be applied to determine how the load is transferred once the gap is eliminated. This concept is important when analyzing load redistribution and resultant support reactions in structures involving intermittent contact.

Compatibility Conditions

Compatibility conditions ensure that the deformations of connected structural elements are consistent with the geometry and boundary constraints of the system. When displacements occur, such as the closure of an initial gap, these conditions are used to relate the deformations to the applied loads and to ensure that the structure deforms without breaking the continuity of the member connections.

Axial Deformation and Elasticity

Axial deformation refers to the change in length of a structural member when subjected to a force along its axis. Elasticity involves the relationship between stress and strain, typically modeled by Hooke's law for linear elastic materials. Understanding these concepts is essential for determining how a member, such as a cylinder or bar, compresses or elongates under an applied load.

Static Equilibrium

This concept requires that the algebraic sum of all forces and moments acting on a system must be zero. In structural analysis, ensuring static equilibrium is critical for solving problems involving support reactions and internal forces. It forms the fundamental basis for analyzing the effects of applied loads and determining unknown reactions at supports.

Transcript

00:01 In this question we have given a solid balls that rolls smoothly from rest and we have to find the horizontal distance the ball travels from point a.

00:12 So first of all i am drawing the diagram for this given cution so this is the track we have given and this is the light height we have given so this height is capital h this is the height small edge and this point is point a we have given and let the ball finally reaches this point and this is given by let b.

00:45 Now we have given data for this question that is capital h is 6 meters small h is given to us that is 2 meter and we know that for solid ball moment of inertia is given by 2 by 5 of mr square.

01:03 Now that this is point 1 and this is at 0 .2 so energy at 1 is given by that is u1 is mgh and kinetic energy at this point is 0 and at this point we can say that final potential energy is mgh and kinetic energy at this point is given by that is half of mv square plus this is due to translational and this is due to rotational so by energy conservation we can say that energy at that total energy at 1 is equal to total energy at 2.

02:03 So at 1 there is kinetic that is potential energy at 1 plus kinetic energy at 1 is equal to potential energy at 2 and kinetic energy at 2.

02:12 Now put all the data in this equation.

02:14 So this is mg h is equal to this is m g into small h plus half of mv square plus this is half of i omega square.

02:28 Now further we can say that this can be written as mgh is equal to this is m gh plus half of mv square plus this is half i is 2 by 5 of mr square and omega is given by v square by r square.

02:43 After solving this we get this is 7 by 10 of v square is equal to this is g into h minus small h finally from here we get the value of v and this will comes out to be 10 by 7 into g capital h minus small h root over.

03:01 Now put all the data in this equation...

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