Derive the equation of motion using the principle of...

Key Concepts

Equation of Motion

An equation of motion mathematically describes how a system evolves dynamically over time. When derived using energy methods, it is obtained by differentiating the total energy (kinetic plus potential) with respect to time, yielding differential equations that govern the system’s behavior.

Potential Energy

Potential energy represents the stored energy of a system due to its position in a force field, such as gravitational or elastic potential energies. Variations in potential energy as the system configuration changes are crucial for deriving the corresponding forces via energy gradients.

Conservation of Energy

This principle states that within an isolated system, the total energy is constant when only conservative forces are doing work. It is used to relate kinetic and potential energies without explicitly invoking the forces involved, simplifying the derivation of equations of motion.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, typically expressed as a function of mass and velocity. In energy methods, it forms one part of the total energy balance and leads to dynamic terms when differentiated with respect to time.

Transcript

00:01 In this question we need to derive the equation of motion using the principle of conservation of energy first in this so kinetic energy we can say that t is equal to 1 divided by 2 multiply by m x squared plus 1 divided by 2 multiply by j now we know that x is equal to r theta right and we because we know that x is equal to r theta and x cap is equal to r theta cap right similarly but we can say that x let's suppose s is a quantity that will be equal to four r theta right because this radius is nothing but four r right so x right so x is equal to 4 r theta that is eventually equal to four times x right because x is equal to r theta only so now from this t is equal to 1 divided by 2 multiply by m r square plus j0 divided by 2 and theta is common so we can say theta square and it is equal to 1 divided by 2 multiply by m r squared plus j not theta square now potential energy that is u you will be equal to 1 divided by 2 multiply by k x is whole square that will be equal to 1 divided by 2 multiply by k 4 x whole square therefore u will be equal to nothing but 8 k r square theta square this is the value of potential energy right now for finding the equation of motion d with respect to d t multiply by t plus u is equal to zero so due to this mr square plus j0 multiply by theta double dash plus 16 kr square k r square theta is equal to zero so this is the equation of motion so in the next diagram we need to again find the equation of motion so let's suppose the kinetic energy in this diagram is equal to 1 divided by 2 mx square right so 1 divided by k equivalent that is the spring constant is equal to 1 divided by 5 times k plus 1 divided by k so there's for k equivalent is equal to 5 times k divided by 6 this is the value of k in the diagram given so now let the displacement of mass be x displacement of mass displacement of mass let's suppose it is nothing but x only so if we find the potential energy u that is equal to 1 divided by 2 multiply by k x squared now k is 5 divided by 6 k multiply by x x is nothing but 2 x whole square plus 1 divided by 2 because in this question we have given 2 pull is so 2 pulle's and the second pulley have the k 2k and multiply by x squared so after solving this we will get some equations that we don't have to solve because we eventually have to find the equation of motion.

04:37 So the condition of equation of motion is d with respect to dt, u plus t should be equal to 0...

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