# How Does Gravity Affect Equations in Vertical SHM Problems?

• brendan3eb
In summary, a mass attached to the end of a vertical spring of spring constant k causes the shm to oscillate back and forth between the equilibrium points. There is an unknown force that suppresses the shm's oscillation.
brendan3eb
I have been doing quite a few SHM problems, and I just have a few questions in general. A lot of questions evolved from one particular problem type: A mass attached to the end of a vertical spring of spring constant k.

My questions:
1. How can we prove that we can use the equation w=(k/m)^(1/2) for this problem. Normally, you can just go:
ma=-ky
a=-k/m * y
a=-w^2*y
y's cancel out
w=(k/m)^(1/2)
but in this case you should have to account for the mg force, but in most solutions, I do not seem mg accounted for?

In one problem, I was asked to solve for the maximum amplitude the shm could have in order to not surpass a certain acceleration. Once again, all answers were along the lines:
ma=-kA
mg=-kA
A=-mg/k
Once again, how can you neglect the mg force?

My only idea is that since we determine the equilibrium point for most of these problems at the beginning - the point where the spring force matches the gravitational force - that they treat this equilibrium point like the spring's equilibrium point and can somehow, magically, neglect the spring force?

I have been doing problems for the last two hours, and still haven't really gotten much further on figuring this out..

you didn't prove that you could use w=(k/m)^(1/2). you have a second order differential equation y'' = -k/m * y
To solve this just try y = a sin (b *t) as a solution and then find out what a and b are.

if you would include an mg force then your new differential equation would become

y'' = -k/m * y - mg. try to prove that if y=F(t) is a solution of the first differential equation, that y = F(t) - mg/k is a solution of the second one.

When the spring is hanging vertically in equilibrium: Tension =Weight i.e ke=mg

When you displace it a small distance,x, Resultant force,F =mg-k(e+x)...use F=ma now.

## 1. What is SHM?

SHM stands for Simple Harmonic Motion. It is a type of periodic motion in which an object oscillates back and forth around a central equilibrium point. This motion is characterized by a constant amplitude and a constant period.

## 2. What are the factors that affect SHM?

The factors that affect SHM are the mass of the object, the stiffness of the spring or restoring force, and the amplitude of the motion. These factors can change the frequency, period, and amplitude of the motion.

## 3. What is the equation for SHM?

The equation for SHM is x = A sin(ωt + φ), where x is the displacement of the object, A is the amplitude, ω is the angular frequency, and φ is the phase constant. This equation is also known as the displacement equation.

## 4. How is SHM different from other types of motion?

SHM is different from other types of motion because it is a periodic motion with a constant amplitude and a constant period. Other types of motion, such as linear motion, do not have these characteristics.

## 5. What are some real-life examples of SHM?

Some real-life examples of SHM include a pendulum swinging, a mass on a spring bouncing up and down, and a guitar string vibrating. SHM can also be seen in the motion of a car's suspension system and the motion of a swing set.

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