Simple Harmonic Motion of a Spring

[ryantosi]

So over the weekend my physics prof has assigned an assignment where one of the questions is as follows and here is my thought process:

A massless spring hangs from the ceiling with a small object attached to its lower end. The object is initially held at rest in a position yi such that the spring is at its rest length. The object is then released from yi and oscillates up and down, with its lowest position being 22 cm below yi. (a) What is the frequency (in Hz) of the oscillation? (b) What is the speed of the object when it is 9.9 cm below the initial position? (c) An object of mass 160 g is attached to the first object, after which the system oscillates with half the original frequency. What is the mass (in kg) of the first object? (d) How far below yi is the new equilibrium (rest) position with both objects attached to the sping?

However, upon inspecting this question I realize that the 22cm stretch is not my maximum amplitude, but rather the distance from my un-stretched spring. There for in order to calculate anything with the one piece of information given, I somehow need to calculate my max amplitude. But in order to do that I need the mass of the object or the spring constant. If anyone could help me out here starting out that would be great. I'm not looking for the answer straight up, but i just would appreciate it if someone could put me in the direction on how to solve the equation because me and my classmates are stumped.

 

[jedishrfu]

what if the 22cm stretch is the maximum? can you solve it then?

 

[ryantosi]

Well could you not manipulate the formula x(t)=Xmcos(wt+∅)? I have been stuck on this question for hours now with no progress at all and am running out of motivation.

 

[jedishrfu]

Im trying to say your assumption was wrong. When the object reached yi - 22cm the force of the spring was enough to counteract gravity and so the object begins to go back up to the yi position and will oscillate between the two positions.

Dont forget that PF rules say you must follow the homework template and show some work before we can help further.

 

[rezeew]

I would suggest approaching this problem by first understanding the concept of simple harmonic motion. This type of motion occurs when a restoring force is proportional to the displacement from equilibrium, and it results in an oscillation around a central point.

In this case, the massless spring provides the restoring force, and the object attached to it experiences simple harmonic motion as it oscillates up and down. The frequency of this oscillation can be calculated using the formula f = 1/T, where T is the period of the oscillation. The period, in turn, can be calculated using the formula T = 2π√(m/k), where m is the mass of the object and k is the spring constant.

Since the object is initially held at rest at a position where the spring is at its rest length, the amplitude of the oscillation is equal to the distance from this initial position to the lowest position (22 cm). Therefore, to calculate the spring constant, we can use the formula k = mg/A, where g is the acceleration due to gravity and A is the amplitude.

Once we have calculated the spring constant, we can use it to solve for the speed of the object at any given position using the formula v = √(k/m)(A^2 - x^2), where x is the displacement from the equilibrium position.

To answer part (c), we can use the concept of resonance, which occurs when the frequency of an external force matches the natural frequency of a system. In this case, the object with mass 160 g is attached to the first object, causing the system to oscillate with half the original frequency. This means that the natural frequency of the system, which depends on the mass of the first object, has also been halved. From this, we can calculate the mass of the first object.

Finally, to determine the new equilibrium position with both objects attached, we can use the concept of equilibrium in simple harmonic motion, which occurs when the net force on the object is zero. This means that the force exerted by the spring must be equal to the weight of the two objects combined. Using this information, we can calculate the new equilibrium position.

In summary, to solve this problem, we need to use the concepts of simple harmonic motion, resonance, and equilibrium, and apply the relevant formulas to calculate the missing variables. I hope this helps you and your classmates in approaching and solving this problem.