# Relationship between elasticity and inertia

### Elastic Section Modulus vs Moment of Inertia | Physics Forums

Newton's First Law (Inertia); Newton's Second Law (Proportionality of force and acceleration); Elasticity (Hooke's Law); Elastic limit The elasticity of the upper thread is evident in the slow pull demonstration, as the References & Links. Inertia is the tendency of an object to resist the change in its motion Elasticity is the ability of a body to resist a distorting influence or deforming force and to return . Momentum is a measure of how motion is in a moving object, and is dependent on the Inertia refers to the resistance of an object to change its state of motion.

Tensile stress is a measure of the deformation that causes stress. Strain is a number without units; therefore, the expression for Young's modulus is If an object of cubic shape has a force applied pushing each face inward, a compressional stress occurs.

Under uniform pressure, the object will contract, and its fractional change in volume V is the compressional strain. The negative sign ensures that B is always a positive number because an increase in pressure causes a decrease in volume.

### Elasticity and Simple Harmonic Motion

Applying a force on the top of an object that is parallel to the surface on which it rests causes a deformation. For example, push the top of a book resting on a tabletop so that the force is parallel to surface. Shear stress is defined as the ratio of the tangential force to the area A of the face being stressed. Units for k are newtons per meter. When a mass is hung on the end of the spring, at equilibrium the downward gravitational force on the mass must be balanced by an upward force due to the spring.

This force is called the restoring force. The negative sign indicates that the direction of the restoring force due to the spring is in the opposite direction from the stretch, or displacement, of the spring.

Simple harmonic motion A mass bouncing up and down on the end of a spring undergoes vibrational motion. The motion of any system whose acceleration is proportional to the negative of displacement is termed simple harmonic motion SHMi.

Certain definitions pertain to SHM: A complete vibration is one down and up motion. The time for one complete vibration is the period, measured in seconds. The frequency is the number of complete vibrations per second and is defined as the reciprocal of the period. The amplitude is the absolute value of the distance from the maximum vertical displacement to the central point of the motion, that is, the greatest distance up or down the mass moves from its initial position.

### Effect of inertia and elasticity on stick-slip motion.

This relationship gives the period in seconds. As in the previous simulation the force is simulated by setting the charge of the particle to a positive value in the Particle Inspector window and giving the strength of an electric field in the positive direction of the x-axis a positive value in the Global Parameters window.

The arrows indicate the directions and amounts of the applied force f white and the resulting acceleration a blue.

The latter two values are indicated numerically in the "Particle Inspector" window. By changing the mass of the particle in the "Particle Inspector" window the simulation shows that the acceleration a and the mass m are inverse proportional.

If you double the mass, the acceleration is divided in halve and vice versa. This relation is valid in general and should be intuitively clear: Such an inverse relation implies, that the product of mass and acceleration is constant. More to this relation will be found in the next chapter.

This relation - the inverse proportionality of m and a - has been tested experimentally and has been found to be valid under all circumstances, at least unless not very high velocities comparable with the speed of light get involved.

This product - m x a - has been taken as definition and as a measure for the strength of a force which leads to the famous equation, set up in the Inertia From experience in daily life we know that an object resists, if we try to make it move faster or if we want to reduce its speed. In traditional physics this resistance is called inertia and is seen as a property of matter. And as a property of matter, it cannot be a Newtonian force. An important and indispensable character of a Newtonian force is that is has to occur as interaction between two objects.

It cannot be the property of a single object. Furthermore inertia can never accelerate an objects. It can only react on acceleration.

And since such an interactive partner is missing, the resistance of a material object against any kind of acceleration is not seen as a force and is not visualized as such in traditional textbooks.

## Effect of inertia and elasticity on stick-slip motion.

At that time, the German physicist Mach proclaimed some principles about general relativity and published the idea that the presence of the matter of all the galaxies in the universe is responsible for the inertial mass of objects here on earth. The broad notion is that "mass there influences inertia here". Until today this idea has never been experimentally approved or disproved and therefore remains speculative.

However it opens the possibility to think of a possible interactive partner for inertia and facilitates to accept the resistance of a material object against acceleration as a real force.

In the following learning material the possibility is offered to accept this idea of inertia as a real force and to visualized it, when ever wanted, as follows. Visualization of a Newtonian force and the equal and oppositely oriented force of inertia as an acceleration-reaction-force By doing so it has to be kept in mind that a force of inertia, even though it is real, is not a Newtonian force, which can be applied to accelerate an object.

The force of inertia is an acceleration-reaction force, which only exists during the process of acceleration. It cannot be applied to an object to cause acceleration. But if a Newtonian force is applied to an object this force of inertia makes acceleration possible by producing the necessary resistance.

Without it the Newtonian force would act against something without resistance, it would pull or push against nothing or emptiness. It seems reasonable to claim that this could have no effect. The following simulations demonstrates different kinds of accelerated movements and allows the user to visualize independently the Newtonian as well as the forces of inertia.

Force of Inertia With the idea of inertia as an acceleration-reaction force the dynamic situation, where an object is accelerated by a Newtonian force can be described as an equilibrium between two forces, a Newtonian force and a force of inertia - an acceleration-reaction force. Both forces can be visualized independently as vectors. How to visualize a Newtonian and an inertial force common for all particle to be set The "Particle Inspector" window offers the possibility to visualize these forces for an individual particle.

How to visualize a Newtonian and an inertial force an acceleration-reaction force for a selected particle An accelerated movement can now be described as if there exists always an equilibrium between an applied Newtonian force and a force of inertia as acceleration-reaction force. This description of a movement is valid for all objects independent a of their mass.

The accordance of the simulation with this last statement can be checked by changing either the mass of the particle or the strength of the applied force by changing the charge of the selected particle in the "Particle Inspector" window. Mathematical description With the idea of inertia as an acceleration-reaction force, the movement of an object being accelerated by an applied Newtonian force can be interpreted as an equilibrium and represented in mathematical form as: An equilibrium between a Newtonian and an acceleration-reaction-force, a force due to inertia, does only exist during the process of acceleration.

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