# What Does the Constant Quantity x(dy/dt) - y(dx/dt) Represent in 2D SHM?

• NeedPhysHelp8
In summary, the physical meaning of x(dy/dt) - y(dx/dt) is that it is the magnitude of the angular momentum divided by the mass. This quantity is conserved both for the motion around an ellipse and for planetary motion.
NeedPhysHelp8
Homework Statement

A particle undergoes simple harmonic motion in both the x and y directions simultaneously. Its x and y coordinates are given by: x=asin(wt) y=bcos(wt) . Show that the quantity x(dy/dt) - y(dx/dt) is constant around ellipse, and what is the physical meaning of this quantity?

The attempt at a solution

Ok so I got the first part of the question:
x(dy/dt) - y(dx/dt) = -abw

Now I have no clue what the meaning of this quantity is? the units I see are m^2/s so is this angular area? someone help please

Think of planetary motion, Kepler's second law.

ehild

Ok great than you ehild! So it's just saying that the same area is covered in equal time all around ellipse.

ehild said:
Think of planetary motion, Kepler's second law.

NeedPhysHelp8 said:
Ok great than you ehild! So it's just saying that the same area is covered in equal time all around ellipse.
That's why this is a bad example. This is not true in this case.

So what's the meaning of x(dy/dt) - y(dx/dt) then? I'm really confused now

What do you think it might mean? The rules of this forum preclude us from telling you directly. You need to show some work.

Well the units of the constant are m^2/s so if that value is conserved around the ellipse this means the area per unit time is constant when traveling around ellipse which makes sense since ellipse is symmetric. I just don't know why you said Kepler's 2nd Law is a bad example it made perfect sense to me. If I'm wrong, can you point me in a better direction about how to think of this problem.

That quantity is constant during the motion, so it is conserved. What conservation laws do you know?

ehild

Oh I just read ahead, gotcha it's conservation of angular momentum! Thanks

Your solution is almost quite correct. Well, the quantity in question is the magnitude of the angular momentum divided by the mass. Anyway, the areal velocity is

dA/dt=1/2 [rxv]=1/2(yvx-xvy)ez,

half the vector product of the position vector with the velocity. This is constant both here and for the orbits of planets. The angular momentum is L=m [r x v]. It is conserved when a body moves in a central force field. Gravity is a central force. The force in your problem is also central, as the acceleration is anti-parallel with the position vector.

ehild

## 1. What is 2D SHM (Simple Harmonic Motion)?

2D SHM, or 2D Simple Harmonic Motion, is a type of motion in which an object oscillates back and forth along two perpendicular axes, following a specific pattern of displacement over time. It is often described using equations of motion such as the ellipse or circular motion equations.

## 2. How is an Ellipse used in 2D SHM?

An ellipse is a geometric shape that is commonly used to describe the path of an object in 2D SHM. In this context, the object's motion is along the major and minor axes of the ellipse, with the object reaching its maximum displacement at the ends of these axes. The equation for an ellipse is often used to describe the object's displacement over time.

## 3. What factors affect the amplitude and period of an object in 2D SHM?

The amplitude and period of an object in 2D SHM can be affected by several factors, including the object's mass, the force acting on the object, and the object's initial displacement. Additionally, the shape and size of the ellipse can also impact the object's motion, as well as any external forces or friction that may be present.

## 4. How is energy conserved in 2D SHM?

In 2D SHM, energy is conserved because the object's motion follows a predictable pattern, with the object continuously moving between potential and kinetic energy. As the object moves towards the center of the ellipse, its potential energy increases, while its kinetic energy decreases. As the object moves towards the ends of the ellipse, the opposite occurs, with potential energy decreasing and kinetic energy increasing.

## 5. What are some real-world examples of 2D SHM?

2D SHM can be observed in various real-world scenarios, such as the motion of a pendulum, the movement of a mass on a spring, or the motion of a satellite in orbit. It can also be seen in the motion of a swinging door or a rocking chair. Understanding 2D SHM can help scientists and engineers design and analyze various systems and structures in the world around us.

• Introductory Physics Homework Help
Replies
4
Views
1K
• Introductory Physics Homework Help
Replies
12
Views
2K
• Introductory Physics Homework Help
Replies
6
Views
4K
• Calculus and Beyond Homework Help
Replies
5
Views
3K
• Mechanics
Replies
21
Views
2K
• Introductory Physics Homework Help
Replies
1
Views
2K
• Introductory Physics Homework Help
Replies
9
Views
988
• Introductory Physics Homework Help
Replies
11
Views
6K
• Differential Equations
Replies
1
Views
5K
• Introductory Physics Homework Help
Replies
4
Views
2K