Finding the Equations of Motion for a Mass-Spring System

In summary, the problem involves a block attached to a square box by 4 springs, placed on a frictionless surface with a mass of ##m##, natural length of ##l##, and strength of ##k##. The position of the block in time is represented by ##x(t)## and ##y(t)##. To find the equations of motion, the distances from each wall to the block on each axis must be calculated using the distance formula. The change in length of each spring and the magnitude of the forces must also be determined, along with the unit vectors in the x and y directions.
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lz975545
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Homework Statement


A block is attached to the sides of a square box by 4 springs. The box is placed horizontally on a frictionless surface (ignore gravity). The mass of the block is ##m##, the natural length of each spring is ##l##, and the strength of each spring is ##k##. Place the block at ##(0,0)##. Let ## x(t), y(t)## the position of the block in time. Find the equations of motion of the block. (Use vectors to break each force up into its vertical and horizontal components)
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Homework Equations



##F = m\frac{d^2x}{dt^2} = -kx##

The Attempt at a Solution


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I'm just a little confused on how to start this problem. Would I use the distance formula from each wall to the block (wall to block to wall) on each axis?
 
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  • #2
lz975545 said:
I'm just a little confused on how to start this problem. Would I use the distance formula from each wall to the block (wall to block to wall) on each axis?
Yes. If the coordinates of the block are (x,y), what are the lengths of the 4 springs? What is the change in length of each spring? What are magnitudes of the 4 forces? In terms of the unit vectors in the x and y directions (and x and y), what are the 4 unit vectors in the directions of the springs?
 

What is a 1 mass 4 spring system?

A 1 mass 4 spring system is a mechanical system that consists of one mass connected to four springs in various configurations. The system can be used to model the behavior of objects that are connected by multiple springs, such as a car suspension system or a building's foundation.

What are the main components of a 1 mass 4 spring system?

The main components of a 1 mass 4 spring system include a mass, four springs, and a fixed support or base. The mass represents the object being modeled, the springs provide the restoring force, and the fixed support holds the system in place.

What factors affect the behavior of a 1 mass 4 spring system?

The behavior of a 1 mass 4 spring system is affected by several factors, including the stiffness of the springs, the mass of the object, and the initial displacement or velocity of the system. Other factors such as damping and external forces can also impact the system's behavior.

How is the motion of a 1 mass 4 spring system described?

The motion of a 1 mass 4 spring system can be described using equations of motion, such as the spring-mass equation or the differential equation of motion. These equations take into account the properties of the system, such as the stiffness of the springs and the mass of the object, to determine the displacement, velocity, and acceleration of the mass over time.

What are some real-world applications of a 1 mass 4 spring system?

A 1 mass 4 spring system has many practical applications, including in engineering and physics. It can be used to model the behavior of car suspensions, building foundations, and even biological systems such as muscles and tendons. The system can also be used to study the effects of vibrations and oscillations in different structures and objects.

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