Equations of motion for disk and spring system

In summary, the problem involves a thin uniform disk with specific properties suspended from a cable line with two springs attached. The equations of motion are solved in terms of x(t) and θ(t), using Newton's 2nd law for rotational and translational inertia. The changes in length of the springs, δ1 and δ2, are related to x(t) and θ(t) respectively. The equations are rearranged to have variables on one side and constants on the other, and the correct expressions for δ1 and δ2 are x-rθ and x+rθ. The final solution involves finding x and θ by solving for the equations of motion and torques.
  • #1
Nikstykal
31
1

Homework Statement


Thin uniform disk with radius r, mass m, and moment of inertia 0.5mr2 is suspended from a cable line where one end is attached to a set point via a spring, and the other end is also attached to a spring but is moving in an upwards direction. Solve for the equations of motion in terms of x(t) and θ(t).
http://imgur.com/xYwVP79

xYwVP79.png

Homework Equations


ΣF=ma, ΣT=Iα, Fs=kδ

The Attempt at a Solution


Used Newtons 2nd law in terms of rotational and translational inertia. δ1 is change in length of left spring, δ2 is change of length of right spring.

Translational: kδ1 + kδ2 - mg = mx"
Rotational: -kδ1r + kδ2r = 0.5mr2θ"

I set δ1 = rθ and then I set δ2 = xIN-x

Substituted in and just rearranged each equation to have variables one side and constant terms on the other. Is that the correct way to work this problem? Thanks
 
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  • #2
just in case anyone comes across this... δ1 is actually x-rθ and δ2 is x+rθ... then ΣF=ma turns into -k(x-rθ) - mg + k(x_in-x-rθ) = mx". do the same for rotation and torques then solve
 

What is the equation of motion for a disk and spring system?

The equation of motion for a disk and spring system is given by F = -kx - bv, where F is the total force acting on the system, k is the spring constant, x is the displacement from equilibrium, and v is the velocity of the disk.

How do you determine the natural frequency of a disk and spring system?

The natural frequency of a disk and spring system can be determined using the equation f = 1/(2π√(m/k)), where m is the mass of the disk and k is the spring constant.

What is the relationship between the amplitude and frequency of a disk and spring system?

The amplitude and frequency of a disk and spring system are inversely proportional. This means that as the frequency increases, the amplitude decreases and vice versa.

How do external forces affect the motion of a disk and spring system?

External forces, such as friction or applied forces, can affect the motion of a disk and spring system by changing the total force acting on the system. This can result in changes in the amplitude and frequency of the system's motion.

Can the equation of motion be used to model other systems besides a disk and spring?

Yes, the equation of motion can be used to model other systems besides a disk and spring, as long as the system can be represented by a simple harmonic oscillator. This includes systems such as pendulums and vibrating strings.

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