Restoring Force- any suggestions?

In summary: So use F=ma to find the acceleration of the mass at the end of its swing. Then use the fact that the acceleration is equal to d^2x/dt^2 where x is distance and t is time. Now you have a DE to solve.In summary, to find the frequency of oscillations perpendicular to the rubber bands in an object connected between two stretched rubber bands at equilibrium, the tension in each rubber band is T and the amplitude is small enough to keep the tension essentially unchanged. Using the small angle approximation and setting the restoring force of the rubber bands equal to the restoring force of a spring, an expression for the frequency can be derived by solving a differential equation.
  • #1
bcjochim07
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Homework Statement


An object of mass m is connected between two stretched rubber bands of length L. The object rests on a frictionless surface. At equilibrium, the tension in each rubber band is T. Find an expression for the freq. of the oscillations perpendicular to the rubber bands. Assume the amplitude is sufficiently small that the magnitude of the tension remains essentially unchanged.


Homework Equations





The Attempt at a Solution



Here's what I tried, but I have no idea if it's right.

I drew a picture, and found that as it oscillates, the tension in each rubber band in the direction perpendicular to the rubber bands is Tsintheta. Using the small angle approximation, I come up with T(d/L) Where d is the distance perpendicular from the original position of the mass. So the total restoring force is 2T(d/L)

and then I set that equal to the restoring force with the spring constant

2T(d/L)=kd and came up with an expression for k and then substuted that into the frequency equation. Any suggestions?
 
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  • #2
your method is completely correct.
Because its simple harmonic motion (i.e. no damping or forcing) all you need to know is the mass (given) and the force - which you found, and looks right.
 
  • #3


I would suggest checking your assumptions and making sure they are accurate. In this case, the assumption that the amplitude is sufficiently small may not hold true in all cases. It would be important to consider the effect of larger amplitudes on the tension in the rubber bands and how it may affect the frequency of oscillations.

Additionally, it may be helpful to consider the energy of the system and how it relates to the frequency. Since the object is resting on a frictionless surface, the energy of the system remains constant and can be used to determine the frequency of oscillations.

Finally, it may be beneficial to explore different scenarios, such as varying the mass of the object or the length of the rubber bands, to see how it affects the frequency of oscillations. This can provide a better understanding of the relationship between the variables and the resulting frequency.

Overall, it is important to carefully consider all factors and assumptions when determining the frequency of oscillations in this system.
 

What is restoring force?

Restoring force is a force that acts in the opposite direction of a displacement from an equilibrium position. It is responsible for restoring an object back to its original position after it has been displaced.

What are some examples of restoring forces?

Some examples of restoring forces include the force of gravity pulling an object back to the ground, the force of a spring returning to its original length after being stretched, and the force of a pendulum swinging back and forth.

How is restoring force related to Hooke's Law?

Restoring force is directly related to Hooke's Law, which states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed. The constant of proportionality is known as the spring constant, and it determines the strength of the restoring force of the spring.

What factors affect the strength of restoring force?

The strength of restoring force is affected by the spring constant, the displacement from equilibrium, and the mass of the object. A higher spring constant, larger displacement, and larger mass will result in a stronger restoring force.

How is restoring force important in everyday life?

Restoring force is important in everyday life because it is responsible for many common phenomena, such as the motion of springs, pendulums, and other oscillating systems. It also plays a crucial role in engineering, as it allows for the design and functionality of various mechanical and electrical systems.

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