Equation of Motion for a Particle Under a Linear Restoring Force

In summary, the problem asks to find the equation of motion for a particle of mass m subject to a force F(x)=-kx, where k is a positive constant. The equation of motion is x''(t)= -kx/m. It is then shown that x(t)=Ceiwt is a solution to this equation for any C, as long as w has one of two possible values: w=√(k/m) or w=-√(k/m). This is derived by taking the derivative of x(t) and plugging it into the equation of motion, which results in -w^2x(t)=-kx(t). Solving for w gives the two possible values.
  • #1
davesface
97
2

Homework Statement


Find the equation of motion for a particle of mass m subject to a force F(x)=-kx where k is a positive constant. Write down the equation of motion as x''(t)=F/m. Then show that x(t)=Ceiwt is a solution to the equation of motion for any C as long as w has one of 2 possible values (i is the imaginary unit, w is omega, t is time). What are those values?

There's more to it, but I am totally lost as to how I can at least start from this information.

Homework Equations


x''(t)=F/m
F(x)=-kx, where k is a positive constant
x(t)=Ceiwt

The Attempt at a Solution


I took the derivative of the last equation listed in b twice to get x'(t)=iwCeiwt and then x''(t)=i2w2Ceiwt, which simplifies to x''(t)=-w2Ceiwt.

I guess that I really would just like to know if I'm anywhere in the ballpark for how the problem should begin. It's not a graded problem, but I hate leaving it unsolved.
 
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  • #2
davesface said:

Homework Statement


Find the equation of motion for a particle of mass m subject to a force F(x)=-kx where k is a positive constant. Write down the equation of motion as x''(t)=F/m. Then show that x(t)=Ceiwt is a solution to the equation of motion for any C as long as w has one of 2 possible values (i is the imaginary unit, w is omega, t is time). What are those values?

There's more to it, but I am totally lost as to how I can at least start from this information.

Homework Equations


x''(t)=F/m
F(x)=-kx, where k is a positive constant
x(t)=Ceiwt

The Attempt at a Solution


I took the derivative of the last equation listed in b twice to get x'(t)=iwCeiwt and then x''(t)=i2w2Ceiwt, which simplifies to x''(t)=-w2Ceiwt.

I guess that I really would just like to know if I'm anywhere in the ballpark for how the problem should begin. It's not a graded problem, but I hate leaving it unsolved.
Looks to me like you are doing the problem backwards! You are first asked to write down the equation of motion. You give as "relevant equations" x"= F/m and F= -kx. Okay, looks to me like the equation of motion is x"= -kx/m.

NOW you can argue that if x= Ceiwt, then x'= Ciweiw and x"= -Cw2eiwt= -w2(Ceiwt which is the same as -kx/m as long as w2= -k/m. That last equation should tell you what values w can have.
 
  • #3
HallsofIvy said:
x"= -Cw2eiwt= -w2(Ceiwt which is the same as -kx/m as long as w2= -k/m. That last equation should tell you what values w can have.

2 questions there:
1. Why is Ceiwt which is the same as -kx/m as long as w2= -k/m?
2. How does w2= -k/m lead me to the values of w?
 

Related to Equation of Motion for a Particle Under a Linear Restoring Force

What is the equation of motion?

The equation of motion is a mathematical formula that describes the motion of an object based on its position, velocity, acceleration, and time. It is typically represented as F=ma, where F is the force acting on the object, m is the mass of the object, and a is its acceleration.

What are the types of equations of motion?

There are three types of equations of motion: constant acceleration equations, uniform acceleration equations, and kinematic equations. Constant acceleration equations are used when an object is moving with a constant acceleration. Uniform acceleration equations are used when the acceleration is changing at a constant rate. Kinematic equations are a set of equations that can be used to solve for the motion of an object in any situation.

How do you solve for the equation of motion?

To solve for the equation of motion, you need to know the values for force, mass, and acceleration of the object. Once you have these values, you can plug them into the equation F=ma and solve for the missing variable. You may also need to use other equations, such as the kinematic equations, to fully solve for the motion of the object.

What are the units for the equation of motion?

The units for the equation of motion depend on the units used for force, mass, and acceleration. In the SI system, the units for force are Newtons (N), the units for mass are kilograms (kg), and the units for acceleration are meters per second squared (m/s^2). Therefore, the units for the equation of motion would be N = kg * m/s^2.

What are some real-life applications of the equation of motion?

The equation of motion has many real-life applications, such as in the design of vehicles and structures, the prediction of projectile motion, and the study of celestial mechanics. It is also used in fields such as physics, engineering, and astronomy to analyze and understand the motion of objects in different situations.

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