Oscillatory motion with a suspended mass

In summary, to find the position of the mass as a function of time, you can use the equation ma = -kx - mg and integrate it twice, considering the initial conditions of the system to determine whether or not oscillatory motion will occur.
  • #1
Jukai
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Homework Statement



A mass m is suspended on a vertical spring. The mass is released from the equilibrium position of the spring without the mass. Find the position of the mass as a function of time, while neglecting friction.

Homework Equations



ma=-kx + mg

The Attempt at a Solution



I set the downward motion as positive, thus explaining my - and + choices in the equation. I posed x=Acos(wt) + Bsin(wt) and integrated the ma=-kx + mg equation twice to get a function of x(t). My main problem is that I don't know what to do with the mg factor. When I integrate I get a gt²/2 factor and that's obviously not oscillatory motion.

If the mass is released from the equilibrium position of the spring without the mass, will the mass simply set a new equilibrium position on the spring without engaging an oscillatory motion? I realize this might be a very simple problem, but I haven't done oscillatory motion in a really long time..
 
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  • #2




It seems like you are on the right track with your solution. To account for the mg factor, you can simply subtract it from both sides of the equation. This will not affect the oscillatory motion, as the gravitational force is constant and does not change with time. So your equation would become ma = -kx - mg, and then you can proceed with your integration as before.

As for your question about the mass setting a new equilibrium position without engaging in oscillatory motion, it ultimately depends on the initial conditions of the system. If the mass is released with a certain amount of initial velocity, it will continue to oscillate around the new equilibrium position. However, if it is released with no initial velocity, it will simply reach the new equilibrium position and remain stationary.
 

FAQ: Oscillatory motion with a suspended mass

What is oscillatory motion with a suspended mass?

Oscillatory motion with a suspended mass refers to the back-and-forth movement of a suspended object around a fixed position. This type of motion is characterized by a constant repetition of a cycle, where the object moves from a starting position, reaches a maximum displacement, then returns to the starting position.

What factors affect the period of oscillatory motion with a suspended mass?

The period of oscillatory motion with a suspended mass is affected by three main factors: the mass of the object, the length of the string or spring, and the force of gravity. The period is longer for heavier objects, longer strings or springs, and a weaker force of gravity.

How is the period of oscillatory motion with a suspended mass calculated?

The period of oscillatory motion with a suspended mass can be calculated using the equation T = 2π√(m/k), where T is the period, m is the mass of the object, and k is the spring constant or the gravitational constant.

What is the difference between simple harmonic motion and oscillatory motion with a suspended mass?

Simple harmonic motion is a type of oscillatory motion where the restoring force is directly proportional to the displacement of the object. On the other hand, oscillatory motion with a suspended mass does not necessarily follow this relationship, as the restoring force may vary depending on the length of the string or spring.

How is the amplitude of oscillatory motion with a suspended mass related to the energy of the system?

The amplitude of oscillatory motion with a suspended mass is directly related to the energy of the system. As the amplitude increases, so does the energy, and vice versa. This is because the amplitude represents the maximum displacement of the object, which is directly proportional to the potential energy of the system.

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