Derive the equation of motion for a particle moving under a central force field

In mechanics, a central force is a force that acts on an object in a direction that is always directed towards or away from a fixed point, known as the center of force. Examples of central forces include gravitational force and electrostatic force. In this context, we will derive the equation of motion for a particle moving under a central force field.

Consider a particle of mass m moving under the influence of a central force field, with the center of force located at the origin of a coordinate system. 

Derive the equation of motion for a particle moving under a central force field-Let r be the distance between the particle and the center of force, and let F(r) be the magnitude of the central force acting on the particle. Let θ be the angle between the position vector r and a fixed reference direction, and let φ be the angle between the projection of r onto the xy-plane and the x-axis.

Using polar coordinates, we can express the position vector r as:

• r = r̂r

where r̂ is the unit vector in the radial direction. The velocity vector v of the particle can be expressed as:

• v = ṙr̂ + rθ̇θ̂

Derive the equation of motion for a particle moving under a central force field-where ṙ and θ̇ are the radial and angular components of the velocity, respectively, and θ̂ is the unit vector in the angular direction.

The acceleration vector a of the particle can be expressed as:

• a = (r̈ - rθ̇^2)r̂ + (2ṙθ̇ + rφ̇^2)θ̂

Derive the equation of motion for a particle moving under a central force field-where r̈ and φ̇ are the radial and angular components of the acceleration, respectively.

Using Newton's second law of motion, we can write:

• F(r)r̂ = ma = (m(r̈ - rθ̇^2))r̂ + (m(2ṙθ̇ + rφ̇^2))θ̂

where F(r)r̂ is the force vector acting on the particle, and ma is the acceleration vector of the particle.

Since the force F(r) is a central force, it is always directed along the radial direction. Therefore, we can write:

• F(r)r̂ = F(r)r̂

where F(r) is the magnitude of the central force.

Substituting this into the equation of motion, we get:

• F(r)r̂ = (m(r̈ - rθ̇^2))r̂ + (m(2ṙθ̇ + rφ̇^2))θ̂

Dividing both sides by m and taking the dot product with θ̂, we get:

• 0 = 2ṙθ̇ + rφ̇^2

Derive the equation of motion for a particle moving under a central force field-This equation tells us that the angular momentum of the particle is conserved. In other words, the particle moves in a plane perpendicular to the direction of the angular momentum, and the magnitude of the angular momentum remains constant.

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Dividing both sides by m and taking the dot product with r̂, we get:

• F(r) = m(r̈ - rθ̇^2)

This equation is the radial component of the equation of motion, and it tells us that the radial acceleration of the particle is determined by the magnitude of the central force and the direction of the velocity vector. It is worth noting that the radial acceleration is always directed towards or away from the center of force.

To simplify this equation, we can use the fact that the angular momentum of the particle is conserved. Using the definition of angular momentum L = mr^2θ̇, we can express θ̇^2 in terms of L and r as:

• θ̇^2 = L^2/(m^2r^4)

Substituting this into the radial component of the equation of motion, we get:

• F(r) = m(r̈ - L^2/(m^2r^3))

Derive the equation of motion for a particle moving under a central force field-This equation is known as the radial equation of motion, and it tells us how the radial acceleration of the particle is related to the central force acting on the particle.

In summary, the equation of motion for a particle moving under a central force field can be derived using Newton's second law of motion and polar coordinates. The resulting equation is the radial equation of motion, which relates the radial acceleration of the particle to the magnitude of the central force acting on the particle. The equation also tells us that the angular momentum of the particle is conserved, and that the particle moves in a plane perpendicular to the direction of the angular momentum.

Conclusion

The equation of motion for a particle moving under a central force field can be derived using polar coordinates and Newton's second law of motion. The resulting equation is the radial equation of motion, which relates the radial acceleration of the particle to the magnitude of the central force acting on the particle. The equation also tells us that the angular momentum of the particle is conserved, and that the particle moves in a plane perpendicular to the direction of the angular momentum. This equation is fundamental to understanding the behavior of objects in a central force field, and it provides a framework for analyzing the motion of planets, satellites, and other celestial bodies under the influence of gravitational force.

FAQ.

Q: What is a central force field?

A: A central force field is a force field in which the force acting on an object is always directed towards or away from a fixed point, known as the center of force. Examples of central force fields include gravitational force and electrostatic force.

Q: What is the equation of motion for a particle moving under a central force field?

A: The equation of motion for a particle moving under a central force field is the radial equation of motion, which relates the radial acceleration of the particle to the magnitude of the central force acting on the particle. The equation also tells us that the angular momentum of the particle is conserved, and that the particle moves in a plane perpendicular to the direction of the angular momentum.

Q: What does the radial equation of motion tell us?

A: The radial equation of motion tells us how the radial acceleration of a particle is related to the magnitude of the central force acting on the particle. It also tells us that the angular momentum of the particle is conserved, and that the particle moves in a plane perpendicular to the direction of the angular momentum.

Q: Why is it important to understand the equation of motion for a particle moving under a central force field?

A: Understanding the equation of motion for a particle moving under a central force field is fundamental to understanding the behavior of objects in the physical world, as it helps explain the motion of planets, satellites, and other celestial bodies under the influence of gravitational force. It also provides a framework for analyzing the motion of particles in other types of central force fields, such as electrostatic force.

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