Oscillating Plane and Z axis - Energy Balance

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• Questionable Thought
In summary, the osculating plane is defined as the plane generated by the Principal unit tangent vector and principal unit normal vector, and the inertial Cartesian z-axis. This plane is often used in the derivation of energy balance equations in elementary fluid mechanics textbooks. The equations for converting force balance into energy balance involve the angle between the z-axis and the normal vector, as well as the differential distances in the direction of motion and normal. Although the z-axis is not always in the osculating plane, the angle between the tangent direction and an arbitrary horizontal axis is the same as the angle between the normal direction and the z-axis. This can also be expressed as a 90-theta angle between two other horizontal axes, which may or may not coincide
Questionable Thought
Hello,

I have a few questions regarding the osculating plane, which I understand to be the plane generated by the Principal unit tangent vector and principal unit normal vector (both are orthogonal), and the inertial Cartesian z-axis.

Ultimately, I plan to understand the geometric/kinematic equations necessary to derive an energy balance of an element starting from a force balance. Also, elementary fluid mechanics textbooks often show this derivation and it seems the z-axis is in the osculating plane.

Here are the equations (image attached). Let the angle between the z-axis in the normal vector be 'theta', let the differential distance in the direction of the motion (tangent vector) be 'ds' and let the differential distance in the direction of normal be 'dn'.

ds*sin(theta) = dz <--- this is the one needed to convert force balance in the z direction into an energy balance.

dn*cos(theta) = dz <---- this one indicates to me that the osculating plane has the z-axis within itI don't really need help with the force balance I just need justification for the geometry.

For instance, is this always/generally true? I feel like the z-axis is not always in the osculating plane. Thus, wouldn't Direction cosines be more general? Thank you

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I will say that generally the z axis is not in the osculating plane. However, the angle, theta, the tangent direction, s, makes with a arbitrary horizontal axis, h1, (that is orthogonal to z axis) is the same angle the normal direction makes with the z axis or equivalently it makes a 90-theta angle with another horizontal axis, h2, (that too is orthogonal to z axis).

These two horizontal axes h1 and h2 are not necessarily x or y, but may coincide with them.

1. What is an oscillating plane and Z axis?

An oscillating plane and Z axis is a system used in physics experiments to study the energy balance of a moving object. The plane refers to a flat surface that moves back and forth in a consistent motion, while the Z axis refers to the vertical axis that the object moves along.

2. How is energy balance measured in this system?

The energy balance in this system is measured by tracking the potential and kinetic energy of the object as it moves along the Z axis. This can be done by using sensors and equations to calculate the energy at different points in the motion.

3. What is the purpose of studying energy balance in this system?

Studying energy balance in this system allows scientists to better understand the relationship between potential and kinetic energy in moving objects. It can also help to determine the efficiency of the system and identify any areas where energy is lost.

4. How does the oscillating motion affect the energy balance of the object?

The oscillating motion adds an additional factor to consider in the energy balance of the object. As the object moves back and forth, its potential and kinetic energy will change, and this must be taken into account in the overall energy balance of the system.

5. What are the practical applications of studying energy balance in this system?

Studying energy balance in this system can have practical applications in fields such as engineering and transportation. It can also help to improve the efficiency of machines and processes that involve oscillating motion, leading to more sustainable and cost-effective solutions.

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