# Motion of particle changing forces

• Pcmath
In summary, the conversation discusses the centripetal force for a particle in uniform circular motion and explores how a force acting towards a specific point can affect the motion of an object. The possibility of using Lagrangian mechanics to derive the equations of motion is also mentioned.

#### Pcmath

So the centripetal force for a particle in uniform circular motion is mv^2/r. This also means that if F > mv^2/r than the particle will get closer to the center and if F < mv^2/r than it will travel further from the center.

Say a 5 kg object at point (0,5) on cartesian plane with initial speed of 10 ms-1 and direction parallel to x-axis, a force of 35 N directed towards point (0,0) acts on the object. It no longer follows a circular path. So is it possible to get a equation that shows the motion of the object? Note the force is towards specific point (0,0) no matter where the object is.

Yes.

Do you know how to integrate?

If the mass has mass m and is at point (x,y), can you write down the force in the x direction and force in the y direction? As a vector if you know about vectors, or two equations if you don't?

Note that I haven't worked through this. We may end up with an integral we can't do.

I tried this method before.

However the problem is that the path is not a circle so it is difficult when working with the angular displacement about the point (0,0) because the force in x and y depends on it.

I think we should end up with a differential equation that can't be solved in terms of elementary functions but I fail to even derive it.

Pcmath said:
Say a 5 kg object at point (0,5) on cartesian plane with initial speed of 10 ms-1 and direction parallel to x-axis, a force of 35 N directed towards point (0,0) acts on the object. It no longer follows a circular path. So is it possible to get a equation that shows the motion of the object?
You will want to approach this problem with Lagrangian mechanics. Simply write down expressions for the kinetic and potential energies in polar coordinates. Then use the standard methods of Lagrangian mechanics to get the equations of motion

The potential energy is ##V=fr ## and the kinetic energy is ##T=\frac{1}{2}m(\dot r^2 + r^2 \dot \theta^2 )##. Everything else is entirely algorithmic to get the equations of motion

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## 1. What is the definition of motion of particle changing forces?

The motion of particle changing forces refers to the movement of a particle or object in response to external forces acting upon it. These forces can cause the particle to accelerate, decelerate, change direction, or remain at a constant velocity.

## 2. What are some examples of forces that can cause a change in motion of a particle?

Some common forces that can cause a change in the motion of a particle include gravity, friction, air resistance, magnetic force, and applied forces such as pushing or pulling.

## 3. How does the magnitude and direction of a force affect the motion of a particle?

The magnitude of a force determines the strength of its effect on the motion of a particle. A larger force will cause a greater change in motion, while a smaller force will cause a smaller change. The direction of the force also plays a crucial role, as it determines the direction in which the particle will move or change its motion.

## 4. What is the relationship between force and acceleration in the motion of a particle?

According to Newton's Second Law of Motion, the acceleration of a particle is directly proportional to the net force acting on it and inversely proportional to its mass. This means that a larger force will result in a greater acceleration, while a smaller force will result in a smaller acceleration.

## 5. How can we mathematically analyze the motion of a particle under changing forces?

To analyze the motion of a particle under changing forces, we can use equations of motion such as Newton's Second Law, which relates force, mass, and acceleration. We can also use vector analysis to break down forces into their components and determine their overall effect on the motion of the particle.