# Is gravity a restorative force in oscillations?

• Elfrid Payton
In summary: So once you sum the forces, the depth of the oscillation is set by the gravity alone?Yes, until someone invents variable gravity! :smile:

#### Elfrid Payton

Gravity is always acting and is always constant, so it acts only to set the initial depth, then all other forces are independent of it. In other words, gravity is not a restorative force, and a restorative force is needed to initiate oscillations.

Perhaps an analogous example explains better: consider a spring, constant k, one end attached so as to be immobile, the other attached to a mass m, the whole thing laying on a horizontal frictionless plane. You pull the spring a distance from its relaxed position and it will oscillate back & forth with frequency sqrt(k/m). Gravity not involved.
Now suspend the spring vertically from the immobile end. The spring will stretch due to gravity pulling on the mass to its equilibrium position. Then you pull the spring down a bit further and again it will oscillate with the same frequency sqrt(k/m). The spring-mass system is a lot easier to analyze. You can include gravity or not in your diff. eq.; you get the same result.

rude man said:
Gravity is always acting and is always constant, so it acts only to set the initial depth, then all other forces are independent of it. In other words, gravity is not a restorative force, and a restorative force is needed to initiate oscillations.

Perhaps an analogous example explains better: consider a spring, constant k, one end attached so as to be immobile, the other attached to a mass m, the whole thing laying on a horizontal frictionless plane. You pull the spring a distance from its relaxed position and it will oscillate back & forth with frequency sqrt(k/m). Gravity not involved.
Now suspend the spring vertically from the immobile end. The spring will stretch due to gravity pulling on the mass to its equilibrium position. Then you pull the spring down a bit further and again it will oscillate with the same frequency sqrt(k/m). The spring-mass system is a lot easier to analyze. You can include gravity or not in your diff. eq.; you get the same result.

Ah, so only restorative forces are included in differential equations for oscillations, and gravity is never a restorative force?

Elfrid Payton said:
Ah, so only restorative forces are included in differential equations for oscillations, and gravity is never a restorative force?
Yes, until someone invents variable gravity!
Actually, here's an example of variable and restorative gravity: suppose you drill a hole thru the Earth passing thru its center, and then drop a rock in at one end. As the rock falls into the hole and thru the center, once it's past the center gravity acts to restore the rock towards the center. One can show fairly easily that the restorative force is proportional to the distance of the rock from the Earth center, and always towards the center, making this a simple harmonic motion. (This assumes uniform-density Earth which of course is not really true, but it makes a dandyexercise!)

But in your (and most) cases, gravity is constant so there's no restorative force coming from it.

Gravity is indeed a restorative force in oscillations. In the equation d2y/dt2+ρgLby/M=0, the term ρgLby/M represents the force of gravity, where ρ is the density of the object, g is the acceleration due to gravity, L is the length of the object, b is the width of the object, and M is the mass of the object. When summing the forces, the force of gravity (Mg) is included in the equation, as it is a force acting on the object. This force is then balanced by the restoring force (ρgLby/M) in order to maintain equilibrium and create oscillations. Therefore, gravity is an essential component of oscillatory motion and can be considered a restorative force in this context.

## 1. What is buoyancy and how does it work?

Buoyancy is the upward force exerted by a fluid on an object that is partially or fully submerged in it. This force is equal to the weight of the fluid that the object displaces. It works because of Archimedes' principle, which states that the buoyant force on an object is equal to the weight of the fluid that the object displaces.

## 2. What factors affect buoyancy?

The factors that affect buoyancy include the density of the fluid, the volume of the object submerged, and the gravitational force acting on the object. The more dense the fluid, the greater the buoyant force. The greater the volume of the object submerged, the greater the buoyant force. And the stronger the gravitational force, the greater the buoyant force.

## 3. What is the difference between positive and negative buoyancy?

Positive buoyancy is when the buoyant force is greater than the weight of the object, causing it to float. Negative buoyancy is when the weight of the object is greater than the buoyant force, causing it to sink. Objects with a density less than the density of the fluid will experience positive buoyancy, while objects with a density greater than the density of the fluid will experience negative buoyancy.

## 4. What is the relationship between buoyancy and density?

Buoyancy and density have an inverse relationship. This means that as the density of an object increases, its buoyant force decreases, and vice versa. For example, an object with a lower density than the fluid it is submerged in will experience a greater buoyant force than an object with a higher density than the fluid.

## 5. How do oscillations affect the buoyancy of an object?

Oscillations, or the back-and-forth motion of an object, do not directly affect the buoyancy of an object. However, the frequency and amplitude of the oscillations can affect the stability of the object in the fluid. If the frequency and amplitude are too great, the object may become unstable and sink due to the increased drag force from the fluid.